There is a huge literature on interpreting probability and this is even larger if you just look at examples of how people use it. A really good place to start is to follow the references mentioned in Hájek's encyclopedia article. A heavily overlapping list including lots of book chapters (among other things) can be found on Branden Fitelson's Probability and Induction course website: Syllabus
Frequency accounts of probability:
Propensity accounts: