Okay, you know how you can't cut support against yourself?
In general, that holds, unless you DISLODGE the unit that's supporting.
Now, if E supports B's hold, then assuming nothing else, that unit is not dislodged, so we go back to the default case of the rule-- support is NOT cut because you can't cut support against yourself.
If it was the other unit, D, that moved, with C's support, then A would cut C's support, and D would cut B's, and nobody moves. And if D moved to A, with support, both A and C would be trying to cut support against themselves, and fail, and it would be a 2v2 move, which fails for both.
Does that make sense?