I believe it's a corruption of the phrase "It begs our belief" - meaning it's not really credible.
May 19 04, 12:04 PM
Stew54
In rather old fashioned usage "beggar" is also a verb (to beggar someone being to reduce them to the status of a beggar). It could be that the phrase "beggars belief" means that something is so incredible it exhausts our capacity for belief.
Taken from:
http://www.funtrivia.com/askft/Question47598.html