Support: Some things exist necessarily, like numbers, and other things depend on something else for their existence, such as rocks and people. A modest version of the principle of sufficient reason. There are arguments against the PSR that there are such things as brute (i.e., inexplicable) facts, and this version allows for there to be such a thing. It only demands that there be explanations for existing things. There is strong support in metaphysics that explicability is the default position and so inexplicability would require strong justification.
Objection #1: Since a contingent thing requires a previous cause, but there was no "previous" before the beginning of the universe, then the universe is the exception to the rule. Response is that this presupposes atheism is true, as it assumes that the universe is all that exists.
Objection #2: While there are no necessary beings, it is necessary that something exists. I.e., while the universe is not a necessary thing, it is necessary that things exist. Response is that his leads to the existence of absurd beings. For instance, if everything that isn't a unicorn does not exist, then this leads to it being necessary that unicorns exist. Furthermore, there doesn't seem to be a logical reason why it should be necessary that contingent things exist in every possible world. There are some possible worlds where no contingent things exist, and so it seems that it isn't necessary that contingent things exist.
2. If the universe has an explanation of its existence, that explanation is God
Support: Many atheists indirectly agree with this premise, as it is the logical contraposition of "if God does not exist, then the universe has no explanation for its existence." This is the typical atheist reply as to why the universe exists: it just exists, inexplicably. Also, the cause of the universe cannot be physical as it came "before" physics existed, so it must be non-physical. But there are only two types of non-physical things: abstract objects and minds. Abstract object can't cause anything, therefore it must be a mind.
Objection: The universe is necessary, not contingent, and thus does not require a causal explanation. Response is that the universe does not fit the characteristics of a necessary thing. It is possible that nothing exists. Or it is possible that different particles exist in the place of the ones we have now. I.e., there is nothing magical about the quarks that make up our universe that make them necessary; there could be different quarks in their place.
3. The universe exists
Support: Obvious. Unless, of course, you're a solipsist.
4. Therefore, the universe has an explanation of its existence
Support: From 1 and 3.
5. Therefore, the explanation of the existence of the universe is God
Support: From 2 and 4.