I've been told that it happens because the surfaces in contact must travel equal distances in any time interval as long as they are in contact. Could someone explain this in a different manner?
It is because there is no such thing as a rigid body. When the two balls collide the molecules on the surfaces of each ball exert forces on each other. Since forces cause accelerations, the molecules move and the balls deform.
Of course it is usually hard to actually see this happening because 1) the collision happens too quickly and 2) some balls are still pretty rigid and the deformation is not substantial. A simple way to see this happening is just to look at your fingers as you push them together (for example, pushing the tips of your first fingers together). You will see that your skin deforms due to the forces the fingers exert on each other. Your skin is not rigid, so deformation occurs.
If the surfaces are in contact ie no relative movement between them, then the surfaces must travel equal distances.
For the surfaces to exert forces on one another the “bonds” behind the surfaces must be compressed - think of them as springs.
Here is a relatively springy golf ball hitting a steel plate so the formation of the steel plate is much less than that of the golf ball.
So although the insides of the colliding objects might travel different distances their surfaces, if the objects do not disintegrate, travel the same distance.
It is true that objects in contact must move the same distance while they are in contact, but its a more difficult way of approaching this.
Start with jello. Rapidly shove two blocks of fresh jello together and you'll watch them deform. They deform because forces between the colliding bits are "local." They're electrostatic forces that try to push the atoms of jello block A away from jello block B. Along the surface where they are in contact, this rapidly leads to the atoms moving the same distance at the same time. This effect occurs on the order of microseconds or nanoseconds. To the naked eye, or even to a high speed camera, it looks like it happen instantly.
However, the rest of the jello block is in motion. It still has momentum. And now the front edge of each block is basically at a standstill. The fact that the front edge is at a standstill is not communicated instantly to the rest of the block. The rest of the block finds out over time. The atoms nearest the contact plane find out first, as the chemical bonds in the jello slowly apply forces to arrest their forward motion. This process takes time, and you can see it with your own naked eye. The whole block of jello wobbles as it transmits these forces and tries to reach a stable configuration.
Every collision has some of this jello like effect. The wobbling happens faster and faster as you get into harder and harder materials (which can transmit the forces through the chemical bonds faster), but it always happens. Here's an excellent video from smarter every day showing it with a golf ball.
All physical effects that you are used to thinking of as "instant" take time in this way. For example, here's a great video of breaking a 1" piece of steel bar by pulling on it. At 28,000 frames per second, it looks instantaneous, but at 150,000 frames per second, you can actually see the bar deforming as it tries to deal with the sudden release of force.