Can you explain why pulling the string down $2x$ raises the pulley $x$?

With a pulley in the configuration of problem 7-25, if you pull the rope down a distance $d$, the total length of rope looping down from the left-hand pulley and then up on the right becomes shorter by $d$. This shortening will be divided equally between the left and the right ropes of the pulley with the mass -- so that pulley moves up by $d/2$.