tldr: Begs the question is a formal term for when a conclusion is not supported by given arguments. Using it the way most people do is technically incorrect, but has become common enough that it is in the gray area where one can consider it the new correct usage.
"Begs the question" refers to a logical fallacy[1], though in modern language usually it's conflated with "invites the question" (which is what people generally mean when they say "begs the question" outside of a comment thread on Reddit).
It's true that people use it that way without issue in everyday conversation, but technically "begging the question" is a logical fallacy in which a statement/proposition presupposes its own truth.
'begging the question' is the name of a specific logical fallacy and is not grammatically correct. You could say 'begs for the question' or as the parent comment suggests...
http://www.quickanddirtytips.com/education/grammar/begs-the-...
tldr: Begs the question is a formal term for when a conclusion is not supported by given arguments. Using it the way most people do is technically incorrect, but has become common enough that it is in the gray area where one can consider it the new correct usage.