Yes, there really is ice underneath the moraine. The surface melt generally flows toward the edges of the glacier, which are lower, and then along the edge once it hits the moraine. The stream along the edge melts the ice and undercuts it, and you end up with a crevasse where the moraine ends. Luckily, the water often seems to find its way underneath the glacier, so the crevasses are not continuous and it's not too hard to find a way onto the ice.