I’ve just been reading about the evolution of the horse.
I see pictures of Hypohippus, with zebra stripes, and Dinohippuswith striped legs.
My question is, how do we know they had those stripes?
You will be delighted to learn, or at least I was delighted to learn, that the technical term for what you’re asking about is “integument reconstruction”. 
Here’s an article about it.
We have photographss, drawings, and preserved bodies of some extinct species like the Passenger Pigeon. For those ancient species there’s a lot guessing involved, and often controversy. But you are supposed to realize it’s just a guess.
tyrannosaurus rex with feathers!
When you get as far back as the dinosaurs, it’s mostly guesswork and “artistic license.” The tendency in recent years has been to depict them brightly-colored, unlike older depictions where they were almost invariably green.
Green is as good a guess as any. The key thing is that the viewer has to realize it’s just a guess, we don’t know what dinoasaurs looked like to that kind of detail.
Even the shape is something of guess because we generally don’t know how much body fat they carried, how much loose skin they had, precise details about posture, or soft tissue features for which no evidence has been found.
Sometimes I wish we could get a version of an artist’s rendering that showed “confidence levels” for each feature. Even the confidence level would be something of an educated guess, but it would help to point out the elements that we’re confident of, and the elements that are the artist’s interpretation.
As far as dinosaurs go, confidence levels are pretty fixed. For species with near-complete skeletons, we know more or less everything about the musculature and bones and just about nothing about the skin texture and coloration.
For species with only partial skeletons known, things get progressively worse.
It’s made up. But we know that lots of horse species have stripes, so drawing an extinct species with stripes is as logical as drawing it with a solid coat.
Coat patterns can vary wildly even between very closely related species. If you just had a skeleton you’d have to be an expert to tell the difference between a lion and a tiger and a leopard, but put some fur on them and they’re wildly different.
The confidence levels would be useful in a number of ways, though. For example, with the issue of feathers on dinosaurs. There are some feathers we’re absolutely certain of thanks to fossilized feathers, or marks on the bone. Other feathers are a reasonable bet given relationships to other dinosaurs. Some feathers (like the “woolly Pachyrhinosaurus” in the linked blog article) are interesting thought experiments, but pretty much fiction given what we know now.
Even on a single dino… some people draw raptors using only the arm feathers we have fossil evidence for, but others give them patches of feathers elsewhere, or even full-body plumage. It’d be interesting to know which features are a virtual certainty and which are merely possible.
We don’t have much to go on for the non-flying dinosaurs with feathers. Quill holes in bones don’t tell us much at all, there could have been spikes sticking out, and nothing resembling a feather. Some of these guys may have had fluffy insulation feathers, they might have looked like a giant Muppet.
So, using this as an example, with mostly made-up numbers, we’d have something like this:
Arm Feathers: 95% confidence of a protofeather or spike. 50% confidence of the size depicted. 10% confidence that it was a protofeather as depicted. 1% confidence that it was red.
Body feathers: 25% confidence of some body plumage. 10% confidence Muppet-like plumage. 1% confidence that it was blue.
To me, anyway, that’s a lot more informative than a picture alone. I can form a general idea of how much is the artist’s interpretation without having to read a thousand pages of literature to recreate the artist’s work.
(And, yes, I know that even the confidences would be subjective and likely to change. The stiff tripod T-Rex of yesteryear wasn’t considered improbable until later work had been done and then we adapted to the model with an upheld tail. But, really, what in science isn’t based on confidence levels derived from current data?)
For some recently-extinct species, there are remains that are well-enough preserved (in tar or in ice) that we can just look and see. And even for the dinosaurs, I think there have been a handful of skin impression fossils found, which would tell you something about skin texture.
cave paintings.
No good for the animals the OP was asking about that lived millions of years ago, though.
Even for animals pictured in prehistoric cave paintings, the colors aren’t necessarily always realistic. Unless a huge percentage of prehistoric mammals happened to be ochre red.
*Og: Why you paint stripes on rhino?
Thog: Me ef’ing with future archeologists! *
As for stripes on early horses, one thing biologists can do is look at what seems to be present in the more primitive extant species as well as what is most common in the infant stages to see what is the most likely basal condition. In horses, stripes on their legs at least seem to be that basal condition. Not sure about the zebra stripes over the entire body, but at least on part of their legs.
Can you explain what a basal condition is, please?
The condition of the ancestor species, before it diversified.
As has already been said, the artist has depicted them that way because many modern members of the horse family have stripes, especially on the legs.
Some scientists have tried to reconstruct the colors of some feathered dinosaurs from pigment molecules preserved in their feathers, but their conclusions have recently been questioned.
Paleoartists generally try to seek modern analogues of fossil animals in order to create their color patterns. In the past large dinosaurs were often depicted as gray or brown, taking large modern mammals like elephants or rhinos as models, or greenish, using crocodiles or turtles. With the present knowledge that birds are descendants of dinosaurs, today artists may use the bright colors of modern birds as analogues.