I was confused about that too, but then I discovered that there are some alternate definitions for the word 'beg' which are (from dictionary.reference.com):

"to fail or refuse to come to grips with; avoid;"

"to take for granted without basis or justification"

I'm not sure whether these definitions of 'beg' or the "begging the question" phrase came first. I'm also not sure which definition of 'beg' came first, and why on Earth the completely separate meanings got merged into one word.