What does it mean to say that a flipped coin has a 50% probability of landing heads?

There are multiple ways to answer this question, depending on what you mean by "probability". This page discusses "subjective probabilities," which are a tool for quantifying your personal uncertainty about the world.

Imagine flipping a coin and slapping it against your wrist. It's already landed either heads or tails. The fact that you don't know whether it's heads or tails is a fact about you, not a fact about the coin. Ignorance is in the mind, not in the world.

So your mind is representing the coin, and you're unsure about which way the coin came up. Those probabilities, represented in your brain, are your subjective probabilities. Probability theory is concerned with the formalization, study, and manipulation of subjective probabilities.

If probabilities are simply subjective mental states, what does it mean to say that probabilities are "good," "correct," "accurate," or "true"? The subjectivist answer, roughly, is that a probability distribution becomes more accurate as it puts more of its probability mass on the true possibility within the set of all possibilities it considers. For more on this see Correspondence visualizations for different interpretations of "probability".

Subjective probabilities, given even a small grain of truth, will become more accurate as they interact with reality and execute Bayesian updates. Your subjective belief about "Is it cloudy today?" is materially represented in your brain, and becomes more accurate as you look up at the sky and causally interact with it: light from the clouds in the sky comes down, enters your retina, is transduced to nerve impulses, processed by your visual cortex, and then your subjective belief about whether it's cloudy becomes more accurate.

'Subjective probability' designates our view of probability as an epistemic state, something inherently in the mind, since reality itself is not uncertain. It doesn't mean 'arbitrary probability' or 'probability that somebody just made up with no connection to reality'. Your belief that it's cloudy outside (or sunny) is a belief, but not an arbitrary or made-up belief. The same can be true about your statement that you think it's 90% likely to be sunny outside, because it was sunny this morning and it's summer, even though you're currently in an interior room and you haven't checked the weather. The outdoors itself is not wavering between sunny and cloudy; but your guess that it's 9 times more likely to be sunny than cloudy is not ungrounded.

Several coherence theorems suggest that classical probabilities are a uniquely good way of quantifying the relative credibility we attach to our guesses; e.g. that even if it's probably sunny, it's still more likely for it to be cloudy outside than for the Moon to be made of green cheese. This in turn says that while the probabilities themselves may exist in our minds, the laws that govern the manipulation and updating of these probabilities are as solid as any other mathematical fact.

For an example of a solid law governing subjective probability, see Arbital's guide to Bayes' rule.