They didn't necessarily symbolize anything. And even when they did, it wouldn't necessarily be the same across the many countries and centuries they were in use. (Generally speaking, older medieval coats of arms were more often just abstract color patterns and things got more intricate over time) Things that were universal ended up having fairly generic meanings; e.g. lions are supposedly brave, strong and noble, so lions were extremely popular. In some cases, stories to explain the arms were invented after-the-fact, as with the War of the Roses story of them picking roses in different colors.

Just to give an example of how trivial the symbolism can be, even when present: There are two European capitals, Berlin and Bern, the origins of whose names are lost, but both happen to sound similar to 'bear' in German (and English). So they have bears as their mascot and on their coats of arms. Not very 'deep'.