When the brakes are checked, what do the firemen actually do? Watching this video (at 06:58 mark), this plane had just made a mayday landing, due to a bird-strike. Because of the right engine being shutdown, it was forced to land immediately, with a full load of fuel. The added weight of the fuel, combined with loss of the additional stopping power of reverse thrust, caused the pilots to bear down on the brakes harder and longer than would be normal in an end-of-flight landing.  

The pilot requests to hold on a taxi-way after vacating the runway, during which time the firemen come up to the plane to "inspect the brakes."  

What do the firemen actually do when inspecting? Are they looking for fire? Glowing brake pads? Do they use an infra-red heat gauge?
There are specific protocols that procedurally are required to be followed. Insofar as brake inspection is concerned, it would have been obvious from the trucks that there was no brake fire. It's also obvious that there's not a major amount of smoke coming from the brakes. As they approached the landing gear, they would have smelled serious brake heating. Glowing brakes would be seen then if not sooner. A close visual inspection for very light smoke and sensing the heat coming off the brakes would give them an idea of the rate of brake vaporization. It's a subjective judgement based on what they see and feel.

As I understand it, tires that dangerously overheat will explode before melting, and that's why thermal plugs are used which melt and deflate the tire to prevent their exploding.

At a landing in Nandi, Fiji (short runway) with what turned out to be a tailwind in a 747-200 at near-max landing weight, we had to use max braking, and I called for huffers to cool things down, and of course for an inspection. I exited the aircraft as quickly as possible to see how bad things were. I was amazed and alarmed to see thick dark grey smoke pouring from the body and wing landing gear wheels. I thought we were close to having a fire. However, the ground personnel assured me that there was no cause for alarm, that the smoke was just brake vaporization that would shortly stop as they blew air on them. It did, and after refueling we departed without incident.