How does conservation of momentum explain how Newton's cradle works?

You lift any number of balls and let go. The moving ball(s) then collide(s) with the stationary ball(s) and causes an equal number of balls to move on the other side.

This demonstrates conservation of momentum, because the number of balls moving before the collision is equal to the number of balls moving after the collision.

Each ball has the same mass, so this equivalent to saying the mass of moving balls remains constant from collision to collision. That takes care of one term. What about velocity? Well, the velocity of the moving balls right before the collision and the velocity of the moving balls on the other side right after the collision are equal. This can be seen, because the moving balls after the collision will never surpass the starting height of the balls on pre-collision side.

So if mass and velocity are unchanging for the system right before and right after collision, then the momentum will necessarily be constant: hence, conserved.

The conservation of momentum explains how Newton's cradle works by demonstrating that momentum is conserved in a closed system. When one ball at rest is struck by another ball, it transfers its momentum to the second ball, causing it to move. The momentum is then transferred through the line of balls until it reaches the last ball, which then swings upward. As the last ball swings upward, it transfers its momentum back down the line of balls, maintaining the overall momentum of the system. This process continues until the energy is dissipated due to friction and other factors.