What it means is that the manufacturer has tested the truck with that amount of load, and determined that it will meet certain performance tests with that weight. Maintain a speed of X mph up a grade of Y%. Accelerate to a speed of X mph in Y seconds. Not overheat in temperatures up to Z degrees.
Unfortunately, the truck makers don't all use the same performance tests, so comparing across brands is difficult. The SAE developed a set of voluntary standards for tow ratings (J2807), but I think
only Toyota uses them so far.
What happens if you tow "heavy"? Ignoring the legal aspects, it means you're on your own as to how well your rig will perform. Some sort of failure (overheating, transmission, tires, bearings, seals, etc.) becomes more likely, but how much more likely becomes just a guess.