How does compatibility work in Uma Musume?

It's the affinity between your trainee and the two parents (and their parents). It shows as symbols on the inheritance screen, the double circle being the best. Higher affinity means the inheritance events trigger more often and pass on more sparks, so a strong parent with bad affinity can be worse than a weaker one with good affinity. Characters who share a lot of race and story links tend to score higher with each other.