The Bayesian interpretation of probability is the interpretation that is (usually) used to justify Bayesian statistics. It bases the use of probability on the belief of rational agents, believing that probability is used to rigorously quantify our uncertainty of various propositions. It is typically contrasted with the frequentist interpretation of probability.
Bayesians treat propositions as random variables, and treat belief in propositions as a probability measure. One begins with a prior belief in a proposition, and then updates it based on the evidence.
There are two major Bayesian schools: Objective Bayesiansism (also called logical probability) and subjective Bayesianism. Objective Bayesians take the view that there is an objectively correct prior to place on a proposition; subjective Bayesians that you can use whatever prior you want. Objective Bayesians typically use Cox’s theorems to define rationality, which posits that the beliefs of a rational agent obey certain probabilistic axioms.
Personally, I think the Bayesian interpretation is total hogwash. It is neither descriptively nor normatively true that people have beliefs that obey the axioms of probability. Indeed, this isn’t even possible, since the sample space isn’t well-defined: you would need to place a probability on everything proposition that could possibly be uttered. Moreover, evidence is quantifiable in small-worlds only, not in large-worlds, so updating your belief in a proposition in a large “based on the evidence”, is a non-starter. It is also circular.