Why does my electric fence arcing cause my LED lights to flicker?
LED fixtures use a tiny fraction of the current of an equivalent incandescent lamp, and it could be that you're seeing induced current from adjacent wires. In other words, the arcing circuit is "transmitting" energy to adjacent wires and the LED circuits are "receiving" enough electrical energy to make some of the LEDs in the arrays glow dimly.

It's possible if you have an AM radio turned on you might hear the arcing fence through the radio from a mile away.

There may be other contributing factors, such as a poor ground connection to the fence charger, or maybe the LED circuits don't isolate both the hot and neutral lines at the circuit breaker. 

Ideally you would have the wiring inspected by a qualified electrician in case there's a safety issue.