First, let me say I'm getting so jealous of all these musically talented people combining technology and music to make amazing videos like this one.
So, this is similar to laying down tracks on top of tracks, or recording one line then another and another and layering them to create the complete song. I used to do this with two tape decks (that's the last time I understood technology enough to do something with layering my own voice like this). The difference here is the professional quality of the sounds and the way the video is edited together to make the whole song come together. Here's another song.
This seems a lot like Pomplamoose's music, but this is a one-man-band and it's a capella (no instruments).