A simple experiment to get an intuitive understanding of pulleys:

Take a piece of string and hold one end in your right hand, then hold your left hand higher and let the string run over it and hang down.

Now as you move your right hand up or down, the free end will move the same distance. But if you move your left hand up or down, the free end must move twice the distance, because you have string on either side of the hand that must both move that distance. So you are amplifying the movement, getting twice the movement at half the force.

If instead you wanted to amplify the force, as in a pulley, then stand on the free end of the string (so it’s no longer free) and pull down with your right hand. You are now amplifying the force exerted on your left hand, because it moves only half the distance of the right, so you get double the force. And this is exactly how a pulley works. Add more loops to get even more force at the cost of even more movement.

I figured this out while playing with the cats, and it made pulleys just make sense. Hopefully it can do the same for someone else :)