I understand why say cats would need a good few to feed the whole litter, and animals with 2 nipples presumably have them due to the inherent symmetry of animals.
But why does a cow have so many? I’m baffled! 
To make it easier to hook them up to milking machines of course.
Maybe some previous ancestor of the cow had multiple offspring so having more than one teat was advantageous. Maybe cows with multiple teats have some other evolutionary advantage, like say having multiple outlets for dispensing nutrients to offspring to save wear on the system, reducing the risk of infection, etc.
Evolution doesn’t design" life forms. Life forms adapt to existing conditions. Perhaps if cows were left alone they would evolve into a form with fewer teats; however, since cows are at this point almost entirely selectively bred by humans and since there is no advantage to humans to breed for fewer udders, the likelihood is that the current udder count will remain unchanged.
That’s what the bulls like. Have you seen Playcow? 
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Why is it that cows have comically large udders but other hooved mammals generally don’t? I.e. horses, donkeys, etc. don’t have particularly noticable udders.
Well, I think goats do, but I can’t recall for sure. Other than that, though, I can’t think of any.
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Cow have large udders becasue they’re the animals we use for milk. If giraffes had large udders we’d be using them instead and you’d have asked the question about giraffes. It’s a variation of the so-called Anthropic Principle. That’s independent of the fact that we’ve selectively bred them for centuries to improve milk production and that’s probably positively correlated with udder size as well.
And yes, goats have similarly large udders for their body size, maybe even larger than typical cows.
Why multiple teats? Beats the heck outta me.
Mostly breeding.
If you ever see a beef cow or buffalo feeding you will see that the udder is much smaller and less noticeable. The exaggerated udder is mostly just a feature of dairy cattle, and is a product of millennia of selective breeding for cattle with large udders. It’s no more natural than the ridiculous ears on a cocker spaniel.
Here is a picture of a lactating beef cow.
http://www.cutlerherefords.com/herefords/females.html
have you seen the size of a calf? it takes a lot of milk to feed one of those.
out of observation i would think that being able to rotate the calf around will help with soreness. those little calfs could teach a hoover a thing or two.
Women are equipped to feed two children at once, thought they usually have one child.
Most likely, the extra teat on cows is to allow the milk glands for one teat some recovery times.
Also, it’s not uncommon for a cow to develop something like mastitis in several quarters, although I don’t know how much of that is the result of cows being bred to produce so much milk in the first place. Sometimes a whole quarter will completly fail. If you ever go to a cattle sale where an older herd is being sold, you’ll probably see quite a few with only three functioning quarters. Nor is it unheard of for a cow to have more than four teats. I’ve seen one that had seven teats. Of course, only four of those produced milk.
So the cow would have lost the extra teats if it wasn’t for human milking?
Surely cows have been around for long enough to evolve before we started all that?
Cows would have retained the extra teats with or without human intervention and no one has suggested otherwise.
As in all evolution it was a combination of both in a self-reinforcing cycle, just as Darwin stated. Finch diets would already have been varied. Mutations led to a fine-tuned ability to exploit a specific part of that diet and that exploitation in turn favoured more fine tuning.
If momma cow’s single teet gets too sore and she pulls away from baby for a few days, maybe baby cow gets to weak and dies.
Wouldn’t take long before cows with 2, 4, 8 teets to rotate their calf on were the only one’s whos offspring survived to breed.
The multiple calves in the past thing could also be the answer, or just part of it.
Well i’ll be gosh darned! 
Have you milked a cow? You don’t milk a single teet. And they don’t stop producing at the same time. When a calf is eating it will move from teet to teet. Likewise, a cow may also take on an extra calf from another mother at times. Just like a wet nurse.