'Beggar' - used as a verb, as it is in this phrase, rather than a noun - dates from the 1500s. It means, as Geofbob says, 'to make a beggar of'. The earliest use of a phrase developing this point was in Shakespeare's 'Antony and Cleopatra' (probably 1607). He wrote: "For her own person it beggared all description", talking of the lady herself. Later versions were 'it beggars compare' and even the children's card-game 'beggar-my-neighbour' which appeared in the 1700s. 'Beggars belief', specifically, obviously came onto the scene at some point along the historic line, together with 'beggars comprehension' and a variety of others.