Backpropagation is a way for a computer to figure out if it got an answer right when it's trying to learn something. It's like practice for a computer: if the computer guesses something wrong, it can use backpropagation to try again and make a different guess.
Imagine a computer was asked to guess how much money somebody earns. It might guess 10 dollars, but that's wrong. With backpropagation, the computer can look at how wrong it was, and then adjust its guess. Maybe next time it'll guess 5 dollars, and that will be closer to the real answer. Backpropagation keeps happening until the computer guesses the right number.