We assume this principle all the time in daily life, science, and so on, and we never witness a counter example to it. If it were false, then we would expect all sorts of things to exist without explanation. But in science, we always assume there is an explanation of something, and we've never seen a counterexample. If one wants to say that the universe is an exception to this rule, then the burden is on him to explain why the rule fails just for universes but not for anything else.
II. If the universe has an explanation of its existence, that explanation is God
This premise is generally agreed upon already by most atheists! The typical answer for an explanation for the existence of the universe is that it doesn't have one; it just exists, inexplicably. So the typical atheist position is: "if God does not exist, then the universe has no explanation of its existence." Which is logically equivalent to: "if the universe has an explanation of its existence, then God exists."
The universe does not exist by necessity of its own nature because there is nothing logically necessary about our universe. The group of quarks that compose our universe could have been replaced with a different group of quarks without logical contradiction. Therefore, if the universe has an explanation of its existence, it must be in an external cause. Something external to the universe would be something spaceless, timeless, immaterial, and supernatural.
III. The universe exists
Hopefully, no controversy here.
IV. Therefore, the universe has an explanation of its existence
This conclusion follows logically from I and III.
V. Therefore, the explanation for the existence of the universe is God
This conclusion follows logically from II and IV: the explanation for the existence of the universe is God.
This is a brief intro. For a more in depth look, see the full article.
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