Use it to express that something is shockingly hard to believe, often due to extreme circumstances. The team's comeback from a five-goal deficit beggars belief.
Pro Tip 2/3
Not Literal
It's a figurative expression, not to be taken literally. The actor's performance was so good, it beggared belief.
Pro Tip 3/3
Negative Connotation
It often carries a negative connotation, implying something is unacceptably extreme. The corruption in the organization beggars belief.