The first screws go back to the Ancient Greeks (or Ancient Romans, depending on where you look for information.) Both Ancient Civilizations would press olives and grapes to make olive oil and wine (still loved today by both the Greeks, Italians and many others) and they used screw presses, a type of machine press in which the ram is driven up and down by a screw...like this.
From Wikipedia's Screw Press Page: the screw shaft can be driven by a handle or a wheel. It works by using a coarse screw to convert the rotation of the handle or drive-wheel into a small downward movement of greater force.
Basically the screw press creates a strong force to be exerted on an object with minimal effort from rotating the handle or wheel - the screw makes it possible.
Thanks to the deliciousness of olive oil and wine, the screw was invented - so thanks grapes & olives!