*Just to explain, Newton's Third Law, which is also commonly called the "Law of Action and Reaction" states that force come in pairs, meaning that, for example, if I were to push against the wall, thus exerting a force on the wall, Newton theorizes that the wall exerts a force of equal magnitude right back onto me.
*This is thought of more practically when you think of the effects of these forces: I push against the wall, but the wall doesn't move, so the wall is thus pushing back onto me, preventing me from going straight through this wall.
*If I were to push onto, say, a sheet of aluminum foil, it would obviously break, but not because it is not exerting a force back, but because the aluminum foil is not strong enough to match my force, so it allows me to go straight through it.
Now applying this law to a diving board:
*If a person were to walk onto a diving board, the diving board gives a little bit (bends) under the person's weight, but still holds up the person. So, the weight of the person is a force pushing down on the board, but since the board does not break under the person's weight, the board is able to match the force the person applies onto it right back to the person. (Just like the example I gave with me pushing against the wall; the wall is strong enough to match my force and not cave in: same with the diving board)
*Then, when the person suddenly exerts a stronger force on the board aimed downward (as if getting ready to jump off of it), the board bends even more than it did before, and this change in force returned by the board is definitely felt by the person standing on it through a trampoline-like feeling; the diving board, in its desire to return back to its equilibrium point (the point in which is doesn't move, typically straight out rather than bent so far down) exerts this force right back to the person (in the direction in which the force of the diving board aims on the person, or the direction exactly opposite the direction of the person's force on the board).
*Then, the person is launched off the board with the same force that he/she initially applied to it. As soon as the force of the person has stopped pushing onto the board, the board flutters back and forth (up & down) until it returns to its equilibrium point, at which it stops moving. Since there is no longer the force of the person acting on the board, the board stops acting on the person and stops moving.
Hope this helps!