The law of conservation of energy states that energy -- the ability to do work -- can't be created or destroyed. Energy can, however, change forms, which the Newton's Cradle takes advantage of -- particularly the conversion of potential energy to kinetic energy and vice versa. Potential energy is energy objects have stored either by virtue of gravity or of their elasticity. Kinetic energy is energy objects have by being in motion.
Let's number the balls one through five. When all five are at rest, each has zero potential energy because they cannot move down any further and zero kinetic energy because they aren't moving. When the first ball is lifted up and out, its kinetic energy remains zero, but its potential energy is greater, because gravity can make it fall. After the ball is released, its potential energy is converted into kinetic energy during its fall because of the work gravity does on it.
When the ball has reached its lowest point, its potential energy is zero, and its kinetic energy is greater. Because energy can't be destroyed, the ball's greatest potential energy is equal to its greatest kinetic energy. When Ball One hits Ball Two, it stops immediately, its kinetic and potential energy back to zero again. But the energy must go somewhere -- into Ball Two.
Ball One's energy is transferred into Ball Two as potential energy as it compresses under the force of the impact. As Ball Two returns to its original shape, it converts its potential energy into kinetic energy again, transferring that energy into Ball Three by compressing it. The ball essentially functions as a spring.
This transfer of energy continues on down the line until it reaches Ball Five, the last in the line. When it returns to its original shape, it doesn't have another ball in line to compress. Instead, its kinetic energy pushes on Ball Four, and so Ball Five swings out. Because of the conservation of energy, Ball Five will have the same amount of kinetic energy as Ball One, and so will swing out with the same speed that Ball One had when it hit.
One falling ball imparts enough energy to move one other ball the same distance it fell at the same velocity it fell. Similarly, two balls impart enough energy to move two balls, and so on.
But why doesn't the ball just bounce back the way it came? Why does the motion continue on in only one direction? That's where momentum comes into play.