In A Nutshell
Incorrectly, writers and speakers often use the phrase “begs the question” to mean “raises the question [of…].” Begging the question, however, has nothing to do with “questions” that you ask. The phrase is actually a logical fallacy. In a debate or argument, if one side sneakily asks the other side to concede the issue being debated, that would be begging the question.
The Whole Bushel
As with just about everything else, the Ancient Romans essentially laid the foundation for modern logical debate (perhaps an oxymoron, but still). They did such a good job that we still use a number of their rules to encourage logical arguments and well-structured debates. We also use their methods of calling out someone debating in an unfair or illogical manner. “Petito principii” from which the English, “begging the question” descends, was one such method.
Imagine a prosecuting lawyer examining a defendant on the stand for beating his wife. The first thing the prosecutor asks the man is, “What color fedora do you wear when you beat your wife?” The whole point of the trial is to determine whether or not that very crime has occurred, but with the wording of his examination of the defendant, the lawyer assumed that the issue at hand (i.e., the “question” of wife-beating) is true.
Any judge (who’s actually awake) would put the kibosh on that line of questioning. The Ancient Romans would have called out “B.S.!” Or, rather “petito principii!”
If you replace the word question with “issue,” it makes more sense. And to beg the question or issue is to ask the opposing side to concede the entire debate, but on the sly, like a lawyer or some other type of talking weasel.
The problem with the modern, corrupted usage is that we already have multiple ways of “raising the question,” but without the shorthand of “begging the question” it becomes rather more difficult to call out the verbal trap laid by our hypothetical lawyer in the example above.