Cantio is a feminine Latin noun meaning 'song'. The plural in the nominative case is 'cantiones'. The adjective 'pius' means holy or religious. The feminine plural form (to match the noun 'cantiones') is 'piae'. So it just means 'religious' or 'holy' songs.

Ordinary people, as you suggest, wouldn't have known much Latin. But most abbeys and monasteries had choir schools and education for some boys was provided. They probably sang these cantiones.
I think people would have understood the sentence 'Christus est natus' as it was declared at the moment of Christmas, (midnight on Christmas Eve).