Specifically, I have a faces specified 'f v1/vt1 ... etc '. I understand that there are 3 vertices specified with respective texture coordinates, but how do those 3 textures end up being rendered onto a triangular face?
You don't have 3 textures per face but 3 points on one texture. Connecting them with lines you'll get a triangle on your texture, and this is the exact part that will be used on your face when rendered.
You may look at this answer.