Essentially my problem is this: his argument relies on the contingency of the Universe. It basically says that because he can conceive of the universe's non-existence, it must be contingent. But the flaw here is that one can only conceive of the non-existence of the universe if you presuppose that it is contingent, thereby assuming that which you are trying to prove.
Furthermore, his statement that "anything that exists has an explanation of its existence" is based on the observation of matter within the universe taking different forms (that obviously appear to be contingent, such as a house or a person). He used himself as an example, saying that without an explanation (his parents meeting), he could have not existed. Unfortunately, all that has come into existence is a particular form the matter that makes up his body has taken - matter is neither created nor destroyed (for the purpose of this argument, the exceptions to this rule make no difference), so he cannot logically say that matter itself needs an explanation or is contingent. If he cannot prove that matter's existence is contingent, with the Universe being essentially all that physically exists, I don't think he's postulated a particularly strong case.
Another similar argument he used was being able to conceive of a different Universe, which makes the same error. He cannot conceive of a different Universe (i.e., one with different "laws") because doing so also assumes that which he is trying to prove - that the Universe is contingent. The laws of the Universe might actually be necessary - he has certainly not proven otherwise.
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