The law of the conservation of momentum is explained by Isaac Newton's laws of motion.
Newton's Third Law of Motion states that every action has an equal and opposite reaction. When two bodies strike, the force experienced by body A from striking body B is the equal to the force experienced by body B from striking body A, only it is in the opposite direction.
Newton's Second Law of Motion defines force as the rate of change of momentum, so given the forces of contact are equal, the momentum of A must change at the same rate as the momentum of B. The mass of the two bodies is unchanged, so if A slows down, then B must speed up by the same rate. Therefore there has been no overall change in momentum.
The law of the conservation of momentum can be see all around us. For example, on a pool table momentum is transferred from the cue to the cue ball, and is then further transferred to another ball when they collide. On each occasion the overall momentum is conserved.
The coupling of cars on a train track is another good example. Before impact one car will be at rest while the other will be moving; after collision both will be moving, but more slowly, as some of the momentum has been transferred from the first car to the second.
Conservation of momentum applies not only to collisions but to any interaction between two or more bodies, such as the explosion that occurs when a bullet is fired from a gun. The recoil -- the force experienced by the gun and the shooter from the bullet being fired -- is equal to the force exerted on the bullet by the gun when the firing pin strikes the bullet to trigger an explosion. As the bullet is much smaller than the shooter and the gun, its subsequent momentum is far greater. The momentum of a large body moving very slowly equals the momentum of a much smaller body moving rapidly, however the total momentum of the system remains zero.
Momentum conservation can be hard to accept, particularly as some collisions result in no noticeable movement. Jumping up and down on the ground is a good example, as the principle states that the Earth must move. It does move, but given the huge difference in mass between a human being and a planet, it does so by an infinitesimally small amount. Similarly, a comet or meteor colliding with the moon not only leaves a crater, but also slightly alters the moon's momentum.