Explain why supersaturated solutions will not dissolve any more solute?

A supersaturated solution will not accept any more solid solute molecules into the dissolved state for the exact same reason that a saturated solution cannot; it is saturated like a sponge that cannot absorb and contain any more water.

The explanation directly involves the concept called equilibrium. Place a handful of sugar in two cups of water and the size of the pile of sugar will diminish only up to a point. The reason this point is called equilibrium is to reflect the facts that the rate of solute atoms leaving (exiting) the pile - dissolving - is equal to the rate of solute atoms rejoining (entering) the pile - crystallizing.

Raising temperature will increase the level of solute concentration needed to create a saturated solution. Decreasing this temperature, without removing solute from the cooled solution, gives a supersaturated solution.