Electrostatic force, eF=kq1q2/r^2
Gravitational force, gF=Gm1m2/r^2
Therefore, you know that both the electorstatic and gravitational forces will increase with decreasing distance (attraction). Since the forces as increasing then you know that your particles/objects should be experiencing a change in speed.
From kinetic energy, KE=1/2mv^2, increasing speed increases KE. The increasing KE is due in part to the PE, which is decreasing (Law of conservation of energy).
So, greater distance means greater PE. If you imagine the distance between the electrons approaching infinity, then, intuitively they will be at such a distance that the effect they have on each other is negligible, therefore the force and PE should be zero. (PE would not approach infinity).
But as they approach each other, KE is increasing at the expense of PE. Therefore at a finite distance from each other the PE will be less than the PE at infinity (which is considered zero). Anything less than zero is negative. Thats why objects that repel have positive PE while objects that attract have negative PE.
EDIT: if you go here:
http://faculty.piercecollege.edu/yamadak/ and look for "Negative potential energy" on the left side of the page then it might help you understand if my explanation is insufficient
