Multicore rendering allows the game to utilise more than one CPU core.
Since I don't work for Valve, I can't tell you exactly what turning multicore rendering on in the Source engine does, however in game development, a game will generally have a game loop, which is responsible for processing user input, running the artificial intelligence, drawing graphics, and playing sound, amongst other things.
While the graphics card will do the bulk of the heavy lifting for the drawing of graphics, part of the process and most of the rest of the loop will rely on processing power from the CPU. By enabling multicore rendering you're allowing the game to utilise multiple processor cores, with each core executing an instruction in parallel. This has the effect that the part of the main game loop that isn't being performed on the graphics card will finish quicker, providing potential increases in your frame rate.
Not all implementations of multicore rendering are equal and there have been many issues in the past where enabling it actually causes problems, but the general idea is that with it on, there are more computing resources available so you should get a higher, more consistent frame rate, and with it off, there are less resources available so you should get a lower frame rate.