The earth rotates once every 24 hrs, whilst the moon takes about 28 days to make a full circle. Your rope would wrap around the earth until it snapped.

If you let the earth end just dangle freely, it would move a little slower than the rotational speed of the earth 465m/s (1040mph or Mach 1.5), and alternate between dragging along the ground and being hoisted up into space, as the Moon moved in its orbit from perigee to apogee

If this was feasible (it isn't), you could grab the cable when the Moon is at it's closest, then hold on and let it pull you into space. Timed right, you'd get a maximum altitude of 42,200km, which is enough to get to geostationary orbit and then some