If you bet ten dollars on a horse to win, then you have ten dollars riding on that horse. Extend that to whatever you like.

It means there is a lot at stake, either money or prestige, depending on an outcome.

"a lot riding on" means that something is very important, and if that fails, many other things will fail because they depend on it. So for example, if you had a lot riding on passing a test, it means that it's very important to pass that test.