It seems counter-intuitive to me that the two measurements eventually become equal even though I understand that they clearly do.
Is there a simple explanation (beyond the math simply adding up) as to why this is the case?
So it’s just a coincidence. Note that the real point of correspondence is probably not exactly -40.00 degrees.
Basically, they have to be the same at some value since two non-parallel lines have to intersect somewhere.
(and by the same logic, they can only have the same value once).
Not really. The 0 point of the Celsius scale is the freezing point of water, the 0 point of Fahrenheit is the freezing point of a specific mixture of brine. (And, as someone very intelligent once noted, is colder than it ever gets in Denmark, eliminating the need for negative numbers in logbooks). So, the 0 point in Fahrenheit is lower than the 0 in Celsius, but since degrees are smaller in Fahrenheit, Celsius eventually catches up. That it does so at a nice round number is just coincidence.
There really isn’t any explanation beyond “the math” - if (by definition) 0C=32F and 100C=212F that’s just where the lines cross, as here:
http://nuclear-imaging.info/site_content/wp-content/uploads/2012/03/fahrenheit_to_celcius.png
When you have two different scales with units of different “sizes”, there will always be one point where they have the same value. For many circumstances, this value is zero because we are making ‘absolute’ measurements. For example, a length of zero is zero in any unit.
The Celsius and Fahrenheit scales, however, do not place their zero points at absolute zero, and they actually place their zero points at different temperatures from each other. Zero Celsius is the freezing point of normal water, but Fahrenheit chose his zero point based on the lowest temperature he could achieve from a salt and water mixture.
As for why they eventually have the same temp, think of it like this: Start from the freezing point of water (0 C and 32 F) and start moving down the scale toward colder temperatures. For every decrease of 1 degree C, you have to decrease by 1.8 F. So after a decrease of 10 C, you have a decrease of 18 F. The temperature is now -10 C or 14 F (because 32-18=14). After a total decrease of 20 C, you have a decrease of 36 F. The temperature is now -20 C or -4 F. The Fahrenheit numbers are dropping at a faster rate, but the Celsius number had a ‘head start’ by being at 0 initially instead of 32. Eventually, the Fahrenheit value will ‘catch up’ with the Celsius value, at -40 degrees.
With Kelvin temperatures, a change of 1 K is equal to a change of 1 C, so those two temperatures will never have the same value, they will always be offset by 273. There is, however, a temperature where the Kelvin temp and the Fahrenheit temp have the same value. Finding this value is left as an exercise for the reader. 
There is also an absolute temperature scale whose degree size is the same as for the F scale, and whose zero matches that of Kelvin (by definition, as both are absolute scales): the Reaumur scale.
To be fair, it’s a little unsettling that the point is exactly -40 and not -39.2817 or +pi or some other value. But since the Fahrenheit scale is already incredibly artificial, it’s not so surprising after all.
No, that’s the Rankine scale. The Reamour scale is an octagesimal scale: the freezing point of water is 0ºRm, and the boiling point is 80º.
Beat me to it. Even in Spain, Reaumur and Rankine aren’t the same thing; one has the same zero point as Celsius and a different degree size, and the other the same zero point as Kelvin and the same degree size as Fahrenheit.
Note that minus forty is an exact equivalence assuming that “freezing point” and “boiling point” mean the same thing for the definition of the two scales: if zero matches 32 and 100 matches 212, then -40 matches -40 and not so much as a nanodegree the more or less.
Why must there be an equivalent point? Given that they don’t both start from absolute zero, the fact that they go up at different rates mean that they must cross over at some point: 100 Fahrenheit is not as hot as 100 Celsius, but -100 Fahrenheit is not as cold as -100 Celsius, so somewhere in between X Fahrenheit must be neither colder nor hotter than X Celsius.
Footnote: Reaumur and Fahrenheit also cross over, of course; the figure is the less friendly but still exact -25.6 degrees.
And he says something rather dubious:
Everybody?
There’s a lot of agreement that one scale would be better than two, and Celsius is the logical candidate (being more widely used). But the Fahrenheit scale has some advantages: The range 0 to 100 covers the typical range of climate experienced by the vast majority of humans. And a difference of one degree F is about the minimum increment that matters.
No - as others have noted, it’s exact.
It’s not that surprising. Fahrenheit chose a scale of 180 degrees between freezing and boiling. The Centigrade/Celsius scale chose 100 degrees of difference between the same two points. When both scales used nice whole numbers for both freezing and boiling points, it became likely that the intersection point between these two non-parallel lines was going to be , if not a whole number itself, then some simply fraction.
To show this, consider what would happen if the relationship between C and F was the approximate formula you can use to quickly transform one to the other without getting bogged down in the math – take the temperature in Celsius, double, and add 30. It gets you within a couple of degrees over the temperature range people usually live in. For that rule, the temperature at which both scales are the same is -30, another nice round number.
They both chose a fixed value for water freezing 0C 32F and a fixed value for boiling at STP - 100C=180F
Therefore the ratio of one degree to the other is 5:9
There’s maybe 1 in 5 odds that the point where they cross would be an integer in both scales. Every fifth C degree is an even F degree.
The joke goes that Fahrenheit wanted a scale anybody could re-create; he was using ice and salt to get the lowest possible temperature that gets you, and labelled that zero. Then he used human body temperature to mark the 100, but all that fiddling with cold gave him a slight fever. 
However, more likely is that he used freezing and lowest possible temperature and then divided the scale into halves over and over until he had 32 divisions. I suspect the scale has been redefined so exactly 212 is the boiling point where maybe it should be like 212.1234 or something.
Here is the thought exercise that the two degrees must match at some point.
Let’s start with the boiling point of water 212F and 100C and travel to absolute zero -459.67F and -273.15C. Since the Fahrenheit scale starts higher and ends lower, at some point they must read the same. Imagine two cars on a straight interstate, one starts 212 miles from the state line and finishes 459.67 miles beyond it. The other car starts 100 miles from the state line and finishes 273.15 miles beyond it. No matter the speed they drive, if they stop for gas or eat ofr have a flat tire, at some point, the cars have to be next to each other.
Mention of -40° always reminds of Jack London’s story when it was colder than -50°F (-45.6°C):
[QUOTE=Jack London in To Build a Fire]
Fifty degrees below zero … did not lead him to meditate upon his frailty as a creature of temperature, and upon man’s frailty in general, able only to live within certain narrow limits of heat and cold; and from there on it did not lead him to the conjectural field of immortality and man’s place in the universe. Fifty degrees below zero stood for a bite of frost that hurt and that must be guarded against by the use of mittens, ear-flaps, warm moccasins, and thick socks. Fifty degrees below zero was to him just precisely fifty degrees below zero. That there should be anything more to it than that was a thought that never entered his head.
As he turned to go on, he spat speculatively. There was a sharp, explosive crackle that startled him. He spat again. And again, in the air, before it could fall to the snow, the spittle crackled. He knew that at fifty below spittle crackled on the snow, but this spittle had crackled in the air. Undoubtedly it was colder than fifty below—how much colder he did not know.
[/QUOTE]
[SPOILER]… He wet himself halfway to the knees before he floundered out to the firm crust. He was angry, and cursed his luck aloud. He had hoped to get into camp with the boys at six o’clock, and this would delay him an hour, for he would have to build a fire and dry out his foot-gear…
He worked slowly and carefully, keenly aware of his danger. Gradually, as the flame grew stronger, he increased the size of the twigs with which he fed it. He squatted in the snow, pulling the twigs out from their entanglement in the brush and feeding directly to the flame. He knew there must be no failure. When it is seventy-five below zero, a man must not fail in his first attempt to build a fire—that is, if his feet are wet. If his feet are dry, and he fails, he can run along the trail for half a mile and restore his circulation. But the circulation of wet and freezing feet cannot be restored by running when it is seventy-five below. No matter how fast he runs, the wet feet will freeze the harder.
[SPOILER]No wind had blown for weeks, and each bough was fully freighted. Each time he had pulled a twig he had communicated a slight agitation to the tree—an imperceptible agitation, so far as he was concerned, but an agitation sufficient to bring about the disaster. High up in the tree one bough capsized its load of snow. This fell on the boughs beneath, capsizing them. This process continued, spreading out and involving the whole tree. It grew like an avalanche, and it descended without warning upon the man and the fire, and the fire was blotted out! Where it had burned was a mantle of fresh and disordered snow.
The man was shocked. It was as though he had just heard his own sentence of death.[/SPOILER][/SPOILER]
This is the correct and simplest answer.
Since we’re on this topic, a little bit of trivia for ya: the freezing temperature of pure water is not exactly 0 °C. It is very close to 0 °C, but it is not exactly 0 °C. In addition, the boiling point of pure water at standard pressure is not exactly 100 °C. It is close to 100 °C, but it is not exactly 100 °C.
Neither of these is quite true (The Master speaks and Wiki article).
Fahrenheit originally had 3 points of reference: colder than it ever got in Denmark (0 degrees), the freezing point of water (roughly 30 degrees), and human body temperature (roughly 90 degrees). Then, Fahrenheit decided to scale each of these old degrees up (to get rid of some pesky fractional values), leading to a freezing point at 32 and a human body temperature of 96 degrees.
It was just kind of coincidence that the boiling point of water happened to be roughly 180 degrees above the freezing mark at 212 degrees.
Of course, those early measurements of both freezing points and human body temperatures were a bit in error, so the scale was later re-defined so that 32 degrees was precisely the freezing point of water and 212 the boiling point of water. And at that point, the 180 degree difference was firmly established. In other words, if human body temperature was different and/or if the thermometers of the day were more precise, the scaling would have been different, which is an interesting concept in itself.
Why Denmark? It sounds as arbitrary as the temperature in the basement of the observatory in Paris that someone, whose name escapes me, choose as zero.
Besides I think that on a cold winter’s day the temperature could well drop below -18 C somewhere in Denmark (also outside of Greenland).
I’d go hunt down my classnotes to find out whether the brainfart was on my part of on Nacho’s, but I tossed them ages ago… thanks for the correction.