The reason this happened is that your 2 hot wires are out of phase with each other. Instead of a neutral or ground wire that pairs with one wire to make it 120v, you have 2 wires that are 120v AC, but essentially in opposite directions to each other.
This Answer explains it a bit, and the pics are useful.
This gives a technical explanation, but it's, well, extremely technical.
What normally happens in a 120v circuit is that you have one "hot" wire supplying 120v, one "neutral", and one ground.
The neutral and ground are essentially the same thing, but don't interchange them, since that's wrong on multiple levels, including the legal and safety levels. (The neutral wire is what's supposed to "bring down" the electricity from the hot wire, while the ground wire is there only in emergencies and shouldn't normally have current running through it.) These two wires are what the hot wire are essentially compared to, which is 0 (zero) volts.
The neutral wire at 0 (zero) volts means that the sine wave of the hot wire is never at a higher differential than 120v from that neutral wire, which is where we get the "120v" designation. This is essentially the 2nd pic on the first link, where the sine wave goes above and below zero by 120 volts.
However, when you are dealing with 240v, you have two hot 120v wires that are out of phase with each other (the first pic on that other Answer), you get a maximum difference of 240 volts, when one "hot" is +120v and the other is -120v.
Let me try this another way, if that wasn't clear, because it probably wasn't due to the jargon.
So, you have a mound of dirt that's 12 feet tall and you want to jump down off it to the sidewalk. This is 120v. Your max fall is 12 ft. You could also jump off the slope of the mound anywhere, but the highest point is 12 feet. The mound is the 120v hot wire and the sidewalk is the neutral wire.
Right next to that mound is a 12 ft deep hole. You can jump down to that from where you're standing on the sidewalk and the drop is still 12 ft. This is what happens when the 120v goes negative. No matter if you jump off the mound to the sidewalk or from the sidewalk to the hole, it's still 12 ft. The hole is part of the hot wire, but at a different point in time.
But, if you try jumping from the mound to the hole, that's a 24 ft drop. This is 240v. If you, as a person, are only prepared to make the 12 ft jump and end up going 24 ft, there will be a disaster. At best, your legs will break (this is the breakers in your electrical box*). At worst, you'll die (in the case of electronics, this will be a fire or massive overload of an appliance, also possibly causing you to die). In this last scenario, the mound is one hot wire and the hole is the other hot wire. The 24 ft jump is because these two wires are out of phase and no neutral wire, there's no middle-sidewalk for you to jump to first, it's just one big jump.
* FYI, every time you trip the breakers, you do damage to them.
Your microwave was expecting the 12 ft jump, not the 24 ft jump, so that's probably fried. At best, there's an internal fuse or breaker that can be replaced, but there's no guarantee other damage didn't happen. And there's no guarantee that fuse/breaker is easy to get to, or that it's designed to even be replaced. Likely, the chip(s) used to deal with the timer, LED display, and buttons has taken a jolt and is fried, meaning that even if some parts of the microwave are ok, the useful bits aren't.