Maybe there's a psychological aspect to language that humans evolved with that allows this to work

From my research there is actually some evidence that speaking prayers in Latin elicits similar effects as doing breathework (Pranayama) does. The words and their cadence can connect you with the divine. And this is all related to how you are breathing when reciting in Latin. The power of the human breathe is really fascinating to study. James Nestor wrote a book called Breath and it is a good introduction into the power of conscious breathing.