Let's say you are throwing a rubber ball against a wall. You know you don't have enough energy to throw it through the wall, so you always expect it to bounce back. Quantum mechanics, however, says that there is a small probability that the ball could go right through the wall (without damaging the wall) and continue its flight on the other side! With something as large as a rubber ball, though, that probability is so small that you could throw the ball for billions of years and never see it go through the wall. But with something as tiny as an electron, tunneling is an everyday occurrence.
Quantum tunneling is possible because of the wave-nature of matter. Confounding as it sounds, in the quantum world, particles often act likes waves of water rather than billiard balls. This means that an electron doesn't exist in a single place at a single time and with a single energy, but rather as a wave of probabilities.