Here's how the argument works:
Premise 1: Everything that begins to exist has a cause.
Premise 2: The universe began to exist.
Conclusion 1: Therefore, the universe must have a cause.
I think that Premise 1 is something everyone agrees upon, although the wording of the premise is pretty important.
- Bertrand Russell and John Stuart Mill based their opposition to the (related) First Mover argument by asking, "Who created God?" But the Kalām argument doesn't deny that there can a non-created Entity; it just argues (in Premise 2) that space-time cannot be entity. If we argued that God created Himself, that'd be nonsense. We don't. He describes Himself as "I AM," suggesting that He simply exists, not that He created Himself. I'll explain that a bit more below.
- If you see a chain of dominoes falling, you realize that they fall because the one before them fell. But you also realize that this cannot extend on for infinity, or they never would have arrived at this point. So premise two is certainly true. So there must be an agent external to the chain of dominoes who flicked the first one.
- Put another way, if you begin counting at negative infinity, you'll never arrive at 2009. Ever. Even in an infinite number of years. So if the universe had begun an infinite number of years ago, it never would have arrived at this point.
- Atheists will argue, "why isn't that true of God as well?" But we don't believe that God existed from negative infinity B.C. onwards. We believe He exists outside of time and that He created space-time. This argument, then, necessarily establishes the Creator as immaterial (that is, not made of matter), immortal, and not bound by time or causality (since causality is a function of space and time). Arguably, it establishes Him as omnipotent as well (since He created the entirity of space-time).
- The Big Bang theory, as it is generally formulated, supports this argument. It says that there is no 13.8 billion years ago. That is, 13.7 billion years ago, time came into being, and that the argument "what came before the Big Bang" is sort of nonsensical, since "before" has temporal implications, like saying, "what happened before there was a before and after?" Or "what domino fell on the first domino?" So for the Big Bang to be the source of all space-time, something or Someone must have caused the Big Bang, and must exist outside of space-time. Existing outside of space-time gets us to those characteristics of God which we described before.
- Arguments for an infinite regression of universes break down as senseless. They're a more complex way of trying to count from negative infinity.
- Lots of atheists argue for a multiverse: they say that the Big Bang created a massive number of universes, which is why we ended up with one so bizarrely well-suited for life. This argument isn't intended against the Kalām argument, but I thought I'd mention that even someone who believed in the multiverse would have to acknowledge that the multiverse began to exist, and thus, needs a cause.
Does this make sense? I think that the usual debate over it makes it needlessly complex, but maybe I'm missing something.