To defy or go beyond what is believable.
'It beggars belief' doesn't refer to the things that beggars believe, although it does relate to begging in a roundabout way. The verb 'to beggar' was coined in the 16th century to mean 'make impoverished; exhaust the means of'. It is found in print in William Roy's religious satire Rede me and be nott wrothe, 1528:
Yerre it be longe so god me mende.
For as sone as the masse is buried
Oure masler shalbe beggered
Of all his ryche possession
To 'be beggared' was to be removed of all that one possessed. The first time that it was used outside of the original 'lose all of one's money' meaning was by that inveterate spinner of nouns into verbs, William Shakespeare, in Antony & Cleopatra, 1616:
For her owne person It beggerd all discription.
Shakespeare invented many new words - 1,631 at the present count - and he also stretched the meaning of numerous existing words. This has given us a long list of verbs that didn't previously exist; for example, he started with the nouns 'champion', 'cow', 'elbow' and so on and used them as verbs. The Bard is generally admired for his inventive use of language, often by the same people who now criticize newly-coined verbs like 'I'll google that and text it to you' ('criticize' itself being considered okay, as it is old).
Most early citations of the figurative use of 'beggared' are of the phrase 'beggar description'. It wasn't until the 19th century that 'beggar' and 'belief' were conjoined. In the religious tract The Scheme and Completion of Prophecy, 1830, John Whitley used 'beggar belief' in describing the thoughts of 'those heathens' who denied the Bible.
Anything that seems so strange as to require a previously held belief to be brought low could be said to 'beggar belief'; for example, the Flat Earth Society, as might be guessed, promotes the view that the earth is flat, like a pancake. Most people would agree that this view beggars belief - the belief in question that is 'beggared', i.e. reduced or impoverished, being the widely held view that the earth is spheroid, like an orange.
See also: the meaning and origin of 'beyond belief'.