Here is a video showing how to measure the speed of light with chocolate, toothpicks, a ruler and a microwave.
Can someone explain how this actually works?
For an explanation, I would suggest reading the info on the link you posted.
The microwave oven essentially forms a resonating cavity for the microwaves, so you get what’s known as a ‘standing wave’ to form within it; such a standing wave has nodes, where the amplitude is zero, and anti-nodes where the amplitude reaches maximum periodically at fixed points. The anti-nodes are where the chocolate melts, nodes leave it unmolten (that’s why usually the plate rotates to guarantee even heating). The distance from one node (or anti-node) to the next is half the wavelength, and wavelength times frequency gives you the propagation speed.
Now… the real trick is proving that the speed of light doesn’t change with the frame of reference.
Just wait half a year and do the experiment again. Though this method is probably not sensitive enough 
ETA: or how about just turning the microwave?
We just discussed this recently.
As interesting as this experiment is, unfortunately you can’t be sure it actually does what it purports to do, i.e. actually measure a physical constant.
To calculate the speed of light using this method, you have to make two other measurements of unknown quantities, but this experiment would have you only measure one, and then trust the word of the microwave manufacturer that the other quantity was measured accurately and correctly.
If the manufacturer actually measured the frequency of their product, you might could trust them. But did they, really? It’s not trivial to measure frequency in the GHz range. It’s much, much easier to just measure the wavelength, and then go look up the speed of light and work the same exact equation the other way to arrive at the frequency. If they did that, all your experiment does is arrive at whatever same value the manufacturer looked up themselves. You haven’t actually proved anything about whatever the real speed of light is.
So how do you measure frequency or wavelength in the GHz range? It probably requires more sophisticated equipment than chocolate bars or marshmallows.
Graham crackers?
You could use a photoelectric effect experiment to determine the energy per photon, and then divide that by Planck’s constant to get the frequency. But then, that’s just pushing the question back to how you measure Planck’s constant, and how you know that E = h*nu. A measurement of Planck’s constant can be made from the shape of the blackbody spectrum, but the relationship between energy and frequency (at least, the way Einstein determined it) requires knowing the frequency, so we’re back where we started.