A high bypass ratio engine provides thrust in two main ways.
The core, which acts the same as a turbojet. This chucks out high velocity, hot air in order to generate thrust. This high flow velocity causes a lot of noise for a small increase in thrust, as was evident in the late 60's as jets got bigger and more powerful.
The FAN part of the turbofan works more like a propeller. This chucks out lower velocity air (relatively cold but still heated due to the work done on it, probably not important to OP), but lots of it to generate thrust. Low flow velocity, less increase in noise for an increase in thrust.
Thrust is, as you may have gathered, a force, and is therefore approximately described as a simple one: Force = Mass x Acceleration
Since the acceleration is across the engine (between the front and the back), and we are only interested in what's going on at the back of the engine, we can say that the thrust is approximated by: Force = Mass x Velocity
From this, we can see that there are two ways to increase the thrust; increase the velocity of the flow or increase the mass of air being chucked out the back. The turbofan uses the fan to increase the mass flow rate, increasing the trust, and giving all the lovely benefits listed in this and the previous posts. Hope this helps!