First of all, the statement in question is NOT a consequence of string theory. It could be, but we don't know the physics of such high energy systems. Accordion to the article "something stringy might happen, we simply don't know".So we will ignore string theory.
The next thing that you have to consider is that Newtons Law for Gravitational Attraction is NOT the best description of Gravity that we have. Currently Gravity is best understood by General Relativity and Einstein Field Equations.
The important difference between these two theories is the fact that in Einstein's Theory, energy and mass are effectively the same and both of them cause gravitational attraction. Saying that both energy and mass cause gravitational attraction is technically incorrect but you have asked to keep the explaination simple.
What Randall (the author of What-If) is saying is that the (electrostatic) energy of the electron moon will be so large that it will create black hole and basically suck all the electrons in it.