the “meat” of one’s hand

Solution 1:

No, meat here means literally flesh.

The author couldn't easily say "... to erase without laying their hand on the work," because laying a hand on something is used idiomatically to mean affecting it even without literally touching one's hand (so poking with a stick eraser would still count). And here, the writer is specifically distinguishing an effect which does require touching one's hand to the work from one that doesn't.

So, the stick eraser allows the erasure to be achieved without resting the side, heel or any other part of one's hand on the surface.

Solution 2:

Oishi-san: I'll just add to @Useless's answer by saying that the distinction is between the fingers of the hand and the bulky (i.e., "meat") parts. The fingers may sometimes be used to blur a pencil line on a drawing, for example, but the bulky part of the hand has no place on the paper.