Thousands of years ago is easy.
Changes are high that you'll have a lot of their descendants living today.
By observing them, noting the differences between de descendants and the earlier versions and making comparisons to other animals alive today you can make excellent educated guessed.

Further more, you might have other types of evidence such as footprints, human drawings of the animal, descriptions,...

And finally, you might even be able to get DNA samples or find partially preserved remains.

If we're talking about animals that lived millions of years ago then we still use the same principles but we just don't have the same abundance of data.
But we could for example extrapolate that a dinosaur and a Casuarius that share similar legs and pelvic design would have walked in a similar fashion. If we then find fossilized imprints of the dinosaur's feet and compare them with those of a Casuarius (stride length, spacing,...) then we might use that to confirm or correct our assumption.

This is of course a very brief description about how we can formulate rational assumptions by combining multiple pieces of information; both ancient and modern. So, just understand that it's much more complex then a brief explanation can do justice.