One sense of "burden of proof" seems to be a game-rule for a (non-Bayesian) adversarial debate game. It is intended to exclude arguments from ignorance, which if permitted would stall the game.

I like this framing, but "burden of proof" is also used in other contexts than arguments from ignorance. For example, two philosophers with opposing views on consciousness might plausibly get stuck in the following dialog:

A: If consciousness is reducible, then the Chinese room thinks, Mary can know red, zombies are impossible, etc.; all these things are so wildly counterintuitive that the position that the burden of proof falls on those who claim that consciousness is reducible.

B: Consciousness being irreducible would go so completely against all the scientific knowledge we have gained about the universe that the burden of proof falls on those who assert that.

Here "who has the burden of proof?" seems to be functioning as a non-Bayesian approximation for "whose position has the lowest prior probability?" The one with the lowest prior probability is the one that should give more evidence (have a higher P(E|H)/P(E)) if they want their hypothesis to prevail; in absence of new evidence, the one with the highest prior wins by default. The problem is that if the arguers have genuinely different priors this leads to stalemate, as in the example.

ETA: tl.dr, what Stabilizer said.