If A cannot move itself, and then starts to move, then there must be a B that is moving it. But now the same reasoning applies to B: if B cannot move itself either, then there is a C that is moving B (which is in turn moving A). And so on. If the same applies to C, then we need to keep going until we come to whatever is moving the whole chain, as each of the other members of the chain are powerless. So we are led to postulate X, which has the power to move without itself needing to be moved by anything else.
Let's look at a more concrete example. This is imperfect, as you will see, but it will serve to drive home the point.
You see a rock on your kitchen table. It begins to move across your kitchen table. But you know that rocks do not have the power of self-locomotion, so something else must be pushing the rock. Indeed, you see that a stick is pushing it. But that doesn't explain the motion of the rock, because you know that sticks don't have the power of locomotion either. So yet a third thing must be pushing the stick. You see a pair of tongs grasping the stick.
So now you know what is pushing the stone? No. Because the same thing applies yet again to the tongs. Tongs can't move themselves either, so something must be pushing the tongs. Indeed, there is a stick tied to the end of the tongs that is pushing them.
Yet again, sticks can't move themselves, so there must be something else pushing the stick.
Somewhere, there must be something that is pushing the chain of sticks and tongs without having to be pushed by anything else. Something that pushes without being pushed. An "unpushed pusher." For a quick and dirty example, we could say that a hand is attached to the stick and is the unpushed pusher, and you now know what is moving the stone, via the sticks and tongs. Of course, the hand isn't really the end of the chain, because it is being pushed by muscles, which are being pushed by motor-neurons, which are being pushed by electrons, and so on. That, however, gets into the argument proper, which we are not interested in quite yet.
Another way of seeing it is by way of a simple computer program.
Or consider a laser light on the wall. It is coming from a mirror. But you know that mirrors can't produce lasers, so it must be coming from somewhere else. And it's another mirror. So you need to keep going until you find where the actual laser is coming from.
This post is simply a concept that needs to be understood before continuing, and hopefully these rough examples will help to intuit the basic idea. In the next post, we finally move on to the existence of God.