Interesting. So in a sense all us mammals & birds are in perpetual state of mild fever trying to defeat fungus and mostly succeeding.
My guess is that our metabolic processes and enzymes and stuff all work best at that temp range, or have evolved to work best at that temp range, and that evolution has reinforced that over time- a higher temp might require more food to keep our larger bodies warm, while cooler temps might not allow us to be as active metabolically, or something like that. And it might have also had something to do with fungus like @kenobi_65 says as well.
Now whether that was specific to humans, or evolved far earlier on the evolutionary chain, I don’t know.
With regards to the benefits of higher temp, note that land animals that can’t regulate their body temperature, like reptiles, often hang out in sunny spots to get up to temperature for their activities. And that fish spoils quicker than meat because enzymes and microbes in fish are adapted to lower temperatures and can’t be slowed down as efficiently by refrigerating as a steak.
In his book Gödel, Escher, Bach: an Eternal Golden Braid, Douglas Hofstadter told the story of a multi-user computer system that seemed to have very poor performance when a certain number of users were online. It was a specific number like 30.
Someone suggested that the problem was easy to fix, just find where the number 30 was stored in the operating system and change it to 1,000.
I had read at some point that mammals have a whole bunch of complex metabolic processes for maintaining body temperature, but the rest of their chemical processes are relatively simple because they only have to work in a tight temperature range.
Reptiles, on the other hand, don’t have complex temperature-maintaining processes, but the rest of their biological chemical processes need to be a lot more complex because they have to operate at a wide range of temperatures.
Wow. Today I learned that:
Thank you.
Indeed. This thread has become a thousand times more interesting than the one I imagined would follow my OP.
So I just gotta ask…
How. How did this happen?
Note the vast range of climates that humans have successfully moved into and lived in for millennia like the Sahara or Australian deserts, glaciated Europe [hi to all my Neander-cousins], tropical rainforest or dry hi altitude desert [yo Denisovans!]. Yes, technology and culture were needed as well, but these have substituted for the need to mess with body temp, and there are relatively few other physiological adaptations once you get past skin colour and oxygen absorption that make living in such climatic extremes necessary.
All that suggests to me that 37C is a perfectly cromulent temperature. It is a sweet spot that works, allowing any selective pressures placed on it from the temperature differential between -20C to +45C to be effectively mediated by human cleverness, not the right genes.
That may be true for other temps, but at some point another temperature would be just too hot and difficult for Inuit while being just right if you were a Kalahari San Goldilocks.
[Yes I recognise this is a teleological argument but the idea of an optimal mean seems to work]
Most proteins begin to fall apart (denature) at around 105º-106º so that’s one reason our body temperatures tend not to be much higher than they are. With severe fevers getting close to that temperature, the body tries to kill off pathogens that are also subject to the same limits of chemistry/biology.
If there are more pathogens at lower temps (like room temperature) than at body temperature, that could explain why we tend to be more hot blooded. It was the most evolutionarily advantageous. Having a very narrow range of normal body temperatures also makes it easier to flush out pathogens with a fever.
Well, when a mummy eukaryotic and daddy eukaryotic love each other very much…
There are plenty of other mammals, such as pumas or deer, that live over a vast range of environmental temperatures from the arctic to deserts, that don’t vary much in their body temperature and manage to survive without technology and culture. In general, mammalian body temperature is adapted to internal physiological needs, rather than being adapted to external temperature. A few mammals, such as camels, will allow their body temperatures to rise in response to heat stress, but these are exceptions.
Oddly Fahrenheit tried to set 100 as body temp, but apparently his personal furnace ran hot or maybe he had a fever.
So, if he had his way 100F would be normal body temp.
Note that Celsius really didnt want his scale named after him, and he originally wanted 0C to be boiling and 100C to be freezing.
So the formula to convert F to C if this were the case would be -5/9(F - 212)?
I guess so. It was changed a year after he died, 1745.
I thought his first name was Daniel.
Childhood nickname. 
It’s a good bet a guy who ended up inventing a temperature scale had an odd childhood. Or at least was an odd child. 
But you can’t just acknowledge that this is teleological and then shrug off that criticism. Human body temperature evolved long before we spread out into such diverse environments - in fact, it evolved before we were human. Evolution has no foresight. Natural selection in the ancestral environment could not have optimized body temperature to the diverse range of environments that future humans might experience.
Chimpanzee body temperature is given as 37.2 C (effectively the same as humans), while that of the Western Lowland Gorilla is 35.5 C, despite living in the same environment. If I had to guess, I would think the lower body temperature of the gorilla is due to its larger size, lower activity, and lower quality diet rather than anything to do with the environment/