"Begging the question" is a technical term in the field of rhetoric. It means making an argument which assumes an unproven point. Let's watch a five-year-old beg the question flawlessly:
"Mom, if you make me go to bed now, you won't get to watch the movie with me!"
True, son, true. But here's the thing: I don't want to watch the movie with you! I want to watch it with Dad! Without you, my dear boy! So your argument has failed on your sweet assumption that I want you around right now. Off you go, XOXO.
Another? We'd better. How about an entirely fictional friend of an entirely fictional homeschooling family (you can tell it's fictional because I'm not a homeschooler):
"But they won't have friends!"
Yes! YES! That's the point! We don't want them to have any friends, at least not the kind they'd find at school. You're a charming person and we appreciate your concern, but you have assumed something about what we want for our children, and it's not an accurate something. PS, have you noticed that our house is swarming with friends for them?
Now, what if a five-year-old boy announced to you that his favorite movie was Saw III? No, wait--The Saint? You would surely want to know what appalling mockery of a parent allowed a five-year-old to watch such an atrocity. But that doesn't mean any question (ie proposition) was begged. A question (ie inquiry) was prompted. Begging the question does NOT mean, "makes one wonder" or "leads one to ask."
Again, a homeschooling parent has great anxiety about homeschooling. Those aware of this anxiety would like to know why these crazy people are homeschooling if they're so unhappy with it. But again, no question (proposition) begged here. Only a question (inquiry) naturally formed in the mind of the hearer in response to the available evidence. (BTW, the polite thing to do here is either to ask or to put the whole thing out of your mind, not pretend to sympathize and then badmouth the anxious homeschooler to your other friend later on.)
Fact: misusing this usage damages the credibility of the [mis]user. The increasing misuse of this usage is, as the homeschoolers know, more evidence that schools do not teach what students need to learn, ie, how to think. As such, it separates the thinkers from the bloggers. So handle with care. "Beg the question" is not an expression or a cliché. When in doubt, look for an actual tired idiom. Or really impress everyone by just saying, "It made me wonder [why anyone cared so much about begging the question]".