Atoms can be (more or less) conceptualized as tiny little spheres with positive charge at the center and a spherical cloud (shell) of negative charges around it. So when two atoms come into contact the electron shells repel each other. (I'll ignore the effects of the redistribution of the electron clouds due to the electromagnetic interaction. If you are interested, search under the term "Van der Waals interaction.") This repulsion is why the ball deforms in the first place.
How then can the ball be picked up? It isn't attraction, it's repulsion again. But taking a step outward to see the surfaces involved gives you the idea a bit better: if you are pressing inward on a sphere your fingers sink into it somewhat. Your fingers will not slide over the surface anymore since they are stuck inside the indentation. It has nothing to do with any sort of an attraction.
Even if you aren't deforming the ball we can still make a similar argument: the surfaces of the ball and your fingers aren't perfectly smooth, so there will always be some places where your fingers are "sunk" into the surface of the ball.