Here is my most concise summary of the contingency argument for God’s existence: Things that don’t have to exist, but do, can only be explained by something that does have to exist.

Here is a version that is more fleshed out:

Things that did not have to exist, but do exist (contingent beings), require an explanation for why they exist, and that explanation must be found in some external cause. If everything that exists had an external cause, however, then there would have to be an infinite number of beings and an infinite regress of causes, and ultimately there would be no explanation for why anything that exists, exists. To explain why things that did not have to exist do exist, there must be at least one being that must exist and cannot not exist. This necessary being has being in Himself, and gives being to all other contingent beings.

And finally, here is a way of cashing out the argument in terms of explaining why everything that exists, exists:

What caused everything to exist? Only things that already exist can cause other things to exist, so the thing that caused everything to exist must be part of the group of all existing things that we are seeking to explain. If every individual thing in the group of all existing things were a contingent being, however, this would lead to an infinite regress of explanation and require that there be an infinite number of existents – none of which have an ultimate explanation. Since an infinite number of things cannot exist and an infinite regress is impossible, not every existent in the group of all existing things can be a contingent being. At least one existent must be a necessary being who has being in himself, and gives being to all other existents.