In British English, beggar is a verb meaning (1) to exceed the limits of, or (2) to impoverish. The first sense is what’s meant in the verb phrase beggar belief, which is used to describe something that exceeds the limits of belief. The less common beggar description indicates that something is difficult to put into words.
This sense of beggar does appear outside British English, though not very often. In American English, beggar is almost exclusively a noun (meaning, of course, someone who begs).
It beggars belief to suppose that trained intelligence officials were unaware of the circumstances in which suspects were detained. [Guardian]
Going beyond that and announcing a further six days of strikes for people who directly look after our vulnerable children simply beggars belief. [quoted in BBC News]
It beggars belief that the GOP is willing to risk national default for the sake of antitax purity. [Newsweek]
What beggars belief is the stupidity of his action. [Herald Sun]