What? How is assuming a conclusion (which is the 'correct' usage of the term) related to the literal meaning of begging a question? I have the answer to that, by the way, and I'd not call it a transparent connection. Here we go:
Why begging the question? Well, petitio (from peto) in this context means "assuming" or "postulating", but it has other (and older) meanings, from which the notion of logical postulate or assumption arose: "requesting, beseeching". So rather than use some fancy Latinate term like postulate or assume, people decided to use the plain English word beg[ging] as a sort of calque for the "requesting" sense of petitio. But even in the 16th century, I think, it was a bit odd to warn people against presupposing the end-point of their argument by telling them not to beg their conclusion.
And why begging the question? The OED's first glosses for question are "A point or topic to be investigated or discussed; a problem, or a matter forming the basis of a problem", and "A subject or proposal to be debated, decided, or voted on in a meeting or deliberative assembly". With these meanings, question more or less fits into Aristotle's warning — it's wrong to "assume the question", i.e. to make an argument that presupposes (our conclusion about) the proposal to be decided. But these days, the word question is much more likely to mean "A sentence worded or expressed so as to elicit information from a person; a query, an enquiry." And in this sense, warning someone not to assume the question either means something quite different (say, not to jump to conclusions about what is being asked), or else it makes no sense at all.
What? How is assuming a conclusion (which is the 'correct' usage of the term) related to the literal meaning of begging a question? I have the answer to that, by the way, and I'd not call it a transparent connection. Here we go: