In your requested layman's terms:
Proving that P=NP would prove that mathematically "hard" problems can be solved "easily." Encryption algorithms are designed around the fact that these hard problems can't be solved quickly by a computer. If P=NP, all modern encryption fails, which means most Internet commerce comes grinding to a halt until another solution can be found.
All NP-complete problems (the "hard" problems mentioned above) derived from the 3-SAT problem, so if 3-SAT were proven to be easily solvable (3-SAT is in P), all NP-complete problems are easily solvable.
This is just one real world application of this problem.