Bypass ducts and spinning blades
Well, let's tackle the second question first: What keeps the fan spinning? It is connected by a shaft to the low pressure turbine and the gasses expanding from the high pressure turbine towards the outside keep it spinning. Look at a good diagram and you'll see two (or three) concentric shafts that connect the low pressure turbine and the fan and the high pressure turbine and the compressor.

As for why bypass, pure jet engines where all the air goes through the combustors are really fuel hungry. You have to have a certain air/fuel ratio to get combustion. You can't lean it beyond a certain point or the fire goes out. Using all the air would use really a lot of fuel. The high-bypass ratio turbofan only takes some of the air through the combustors and just sucks the rest in with the fan and blows is out the back. The difference between that and a turbo-prop is mostly the cowling around the fan.