Greetings!
TD probably explained it but I will give it a try too.
Today we might not think in these terms but the idea is that if someone begs for something they want us to give it to them for FREE.
The "question" is just another way of saying "the point" or premise.
So to say that something "begs the question" means that the foregoing (something) asked for the concession of the point for free (that is without going through the work of proving it).
Often it is ok to concede a preliminary point or argument but usually if someone raises the objection that it "begs the question" they are saying that the person assumed the point without proof (for free) and that the omission creates a flaw in the overall argument. In other words they just glossed over that part of the entire argument, either out of laziness or because the point couldn't be supported or proved.
Frankly, few people in normal conversation use the phrase in this proper and traditional way.
As you noted, modernly, when someone uses the phrase they are really just setting up a further question or point in their exposition. In this sense their use is rhetorical.
People do this all the time and should really say instead "that leads to the unanswered question" or "that leads to the next question" or something similar but that is an uphill battle.
This second usage is now what is common and is now more or less acceptable.
-Eduardo Leaton Jr., Esq.