the thermocouple wires are made of strange alloys. You might think there is a temperature sensor on the end, but it is just the two ends of the wires joined together. I imagine they are welded, but you can actually join them yourself by twisting the bare wires ends together and bashing the joint with a big hammer.
The junction of the two wires generates a tiny voltage dependent on temperature. The multimeter has a scale which interprets this voltage according to some performance curve, and you can read out the temp on the display. They are not always perfectly accurate, it is worth testing by immersing the sensor end in a pan of boiling water to see if it reads about 100C.
K type is the most common, AFAIK the plug and socket are always coloured yellow to identify them. There are others which are more accurate at different temperature ranges. "K" is Ok for fridges, heating radiators, domestic stuff.
Depending on what you are using them for, it can be cost-effective to clip and clamp them in position on whatever you are testing, with the plugs handily accessible, so that you can leave themn in place ready for when you have to come back and test again. They are cheaper bought in bulk.