The problem is as follows. I give you a sequence of coin flips, for example TTTHHTTT, and your task is to determine whether this coin is fair. Obviously you can't answer this question with certainty.
The frequentist approach seems rather ridiculous to me. They pretend to be able to answer this question without knowing anything about coins. Instead of making the assumptions up front, they hide the assumptions in the method of determining whether the coin is fair. For example one would think that to decide whether a coin I give you is fair, you'd need some idea about what kind of coins I will give you. If we were talking about reality, you would be able to say "no it's not fair" without even looking at the data sequence, because in reality there are no fair coins. If we are in a different universe where fair coins do exist, you'd need some idea how many are fair. So obviously whether a coin is fair depends on which universe you live in, but the frequentist method is not parameterized by the universe. The assumptions are implicitly made inside the method.
The bayesian approach asks you first to state your prior ideas about coins. Then you give the bayesian your data, and he will compute for you the probability that this coin is fair. This cleanly separates the correct mathematical derivation from the subjective assumptions, instead of hiding these assumptions inside the method.
Moreover, the bayesian approach is automatic, in principle. Once you make your assumptions clear, the rest is just mechanical derivation. The frequentist approach requires divine inspiration, and is very easy to get wrong. For example you're not allowed to look at the data before formulating your hypothesis. Of course nobody does this right in practice. I've often seen people cherry pick a frequentist statistical test that proves their hypothesis. Given any data and any hypothesis, I can devise a correct frequentist statistical test that proves the hypothesis with arbitrarily large confidence.