How do movable pulleys increase mechanical force?

"It works the same way as a lever ... but you'll have to draw a series of diagrams to see this. If the mechanical advantage is 2x then you also have to pull the rope twice as far to raise the block the same height. Without the pulley - if you pull the rope 1m, the rope holding the weight gets shorter by 1m, and the weight goes up 1m, and you do mg in work. If you loop the rope around a pulley on the weight, then if you pull the rope 1m, the rope gets shorter by 0.5 m on each side of the pulley, and so the weight goes up 0.5m and you have done mg/2 in work, moving the same distance ... so the force must be half."

Reference https://www.physicsforums.com/threads/how-a-movable-pulley-increases-mechanical-advantage.669733/

hope this helps you!