When something burns with a flame, electrons are torn from their atoms as the atoms rearrange to form new molecules. Usually they get re-captured by the molecules, and this is one of the reasons why flames glow — the electrons emit light as they lose energy spiraling in from their paths free through the air to being caught in orbits in the new molecules.

A microwave’s job is to set up a standing wave of electric and magnetic fields within a metal box. The electric fields alternately push and pull electrons left and right, or up and down. In a partially conducting material, the current that sloshes back and forth can heat up an object resistively. Even if the material does not conduct dc electricity at all, if it contains water molecules, their electric polarization directions flip back and forth with the field, making them jiggle and get hot.

If electrons are floating around freely, even for a very short amount of time, they can be shoved far away from their point of origin by the electric field. And then shoved back. And then forwards again. As they move back and forth, they crash into air molecules in the oven, and can knock electrons in them to higher-energy orbits. Then these electrons fall back, emitting light. That’s why you have a glowing blob of plasma over your flame. This plasma is hotter than the rest of the air, and so it tends to rise up to the top of your bowl.

I think they arrange the strength of the microwaves in ovens so that the back-and-forth motion of the electrons in a plasma that gets formed is not sufficient to knock other electrons free from the air molecules. If this were the case, even a small spark somewhere on a piece of food would eventually cause the whole oven to fill with plasma.

The reason the thing oscillates at 120 Hz has to do with how the microwaves are generated and shaped in the oven. Microwaves have a resonant cavity called a magnetron which resonates at a few billion Hz. Left to itself, the microwaves quickly dissipate (the energy goes into your food or gets dissipated in the resistance of the walls). The magnetron is constantly fed more energy from the electrical supply which plugs into the wall. Every cycle of power from the wall puts energy into the microwave cavity twice (a typical nonlinear circuit like a rectifier will make high-frequency noise twice per wall-power cycle — the actual circuit of a microwave is probably more optimized to generate energy in the GHz range but to do it only on two places in the wall power cycle). Then the strength of the microwaves in the oven varies at 120 Hz.

The other reason it could oscillate at 120 Hz is that some microwaves have a metal "fan" on top which spins around on a shaft attached to a motor which runs off the wall current. Rather than cool anything off, this "fan" changes the shape of the metal side of the box by having irregularly shaped fan blades which constantly move. The microwaves make a standing wave pattern inside the oven, but the actual locations of the peaks and troughs of the standing wave depend on the shape of the box. By putting this "fan" in there, the peaks and troughs can be moved around — so as not to burn spots of your food while leaving other spots frozen solid, a common problem with microwaves. If the fields change at about 120 Hz (not surprising given that the motor spins at a multiple of the line frequency), it can make your plasma oscillate like that.

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