There are all sorts of reasons why singing has been unaccompanied in the past - unavailability of instruments or people to play them, communal singing or activities (work, marching) making accompaniment unnecessary, singing seen as an extension of story-telling, the need to focus on the words of a song (which can be undermined by an accompaniment), the fact that the voice is portable and always available wherever you go, to list just a few.

Today, people are much more likely to have the option of accompaniment - if you can't afford or can't play an instrument, you can probably find someone to accompany you if that's what you want. But many people still choose not to have an accompaniment, for some of the other reasons given above. If the song has an important story to tell and you really want people to listen to the words, unaccompanied singing can have a much greater impact. Also, some songs are best sung in a loose, flowing rhythm that defies accompaniment.

I think you'll find that in relgious observance, the voice, using singing to carry the words, is sometimes seen as a pure expression of the soul, whereas instrumental music and accompaniment can be seen as a dangerous distraction from worship. In the UK, the "higher" the church is (taking RC as the high point), the more likely it is to use lavish musical arrangments, sung masses etc, whereas the "lower" churches (e.g. Methodism, Baptists) will tend to use just a simple organ accompanimentexcept for special occasions, and try not to distract from the words being sung. Unaccompanied singing is just a step further in this direction.

That's a bit of a generalisaion, and just an idea to start you off, not a fully-fledged theory.

Marje