How many hairy crazy ants does it take to short out an electrical system?

If the conductors are 1 ants length apart, then one ant is all it takes. If they are 10 ants lengths apart, then 10 ants, if they go top-to-tail.

In reality it will take more as they move around lots. Also, it is probable that it would be a gradual build-up of dead ants. As an ant gets electrocuted it will curl up and/or explode. After a while, the bits of dead ant will eventually bridge the circuit.

As for why... well, who knows what goes through the mind of an ant? (besides 110V)

It is well known and documented that mice chew through cables because of the 50/60Hz buzz they produce. Maybe the ants are attracted by the EMF exciting certain areas of their tiny minds?

I had it happen to me :-) !!!
Really.
I may even have photos somewhere.

A junction box in my daughter's room blew a fuse / breaker.
From memory this was within about the last 5 years

Inspection showed that the box was largely filled behind the switch with a vast quantity of ant bodies and that major corrosion had occurred to the switch. I may have had to replace the junction box.

Crushed ants produce Formic Acid and presumably this plus a nice warm dry place + enough bodies is trouble. I don't know if they built a nest in the box proper (I suspect so) or used it as a graveyard.

Ants are probably not much of a issue for low voltage circuits. I expect their body impedance to be fairly high, seeing as they have exoskeletons and no sweat glands. However at high enough voltage it could be a problem. I don't think a individual ant getting fried by 220V is much of a issue to the circuit itself. The current draw should be minimal, and the ant should burn up quicky enough.

There are two real dangers: First as Majenko said, dead ant bodies could build up eventually making a more permanent connection that could be a significant fire hazard. Second, a ant burning up could possibly catch something else on fire.

As to why ants are attracted to eletrical boxes, I don't know. I'm not sure they are in the first place. I certainly haven't seen any evidence of that in my house, but I don't store my sugar in the electrical boxes. Some animals can sense electric fields, although those are usually aquatic. For example, this has been shown in sharks and platypuses. I have no idea whether is true for ants. Again, I would check the original assumption that ants are attracted to electricity in the first place.