Notice the complete reasoning in the link you give:
(12) A: "The drinking age should be lowered to 18, because
18-year-olds are mature enough to drink."
Context: B's reason for disputing the conclusion is the same as [the] reason for disputing the premise here.
So, the problem is, B already agrees that people who are mature enough to drink should be able to drink legally; what she disagrees is that 18-year-olds are mature enough to drink. Unless she can be convinced of the latter, this begs the question, "but why do you think 18-year-olds are mature enough to drink"?
(So, another argument, like for instance, "driving age is 18; don't you think that driving requires more maturity than drinking?" is necessary to sway B.)
Notice that the "context" could be different:
Alternate Context: B's reason for disputing the conclusion is different from her reason for disputing the premise here.
Here B agrees that 18-year-olds are mature enough to drink, but she has a different reason for opposing the lowering of drinking age (perhaps she thinks alcohol should be forbidden for all ages, and sees a 21 year age of drinking as a lesser evil than a 18 year age of drinking (and yes, this is a reasonable position; check your own opinion about heroin, for instance)). In this case, the argument would not be begging the question (or that precise question, anyway).