This argument attacks the concept of a Creator-God, or more generally Creation ex nihilo. It doesn't purpot to disprove the God-Creator in the banal sense, but rather to refute the idea that God is the Creator of Reality. Instead, God is delegated to an artificer at best, a part of Existence rather than what is underlying it. This reduced entity may be considered in awe, but does not provide an ultimate answer to questions such as why are things the way they are, why is there something instead of nothing, and so on.
Let us assume, for the sake of argument, that God exists.
The argument is founded on the assumption that there is one, lowest, level of Existence which is truly Real - this is assumed to contain God, if He exists. In the usual conception, God then creates another element of existence, Creation, seperate from himself. There are two options for this creation - either it is a part of the fundamental, Real, existence, or its existence is a seperate existence.
In the first case, God creates the world much like we might create a virtual world on a computer - by assembling the elements of his layer of existence to form the world. This layer may consist solely of God's thoughts, for example - there is no need to assume a dichotomy between God and other things existing in the underlying reality. It would be inaccurate and misleading, then, to say that the created world is truly Created (ex nihilo) - rather, it is merely Assembled. Any difference in the virtual world created by us must correspond to a real difference in the underlying computer hardware; in the same way, any difference in Creation must correspond to a real difference in the underlying Real layer of existence. The "created" world is not really created; it is more like a distorted, partial reflection of the underlying layer's reality. This god, then, does not provide an underlying explanation to why things in Creation are the way they are, since any difference in Creation reflects a difference in God's-own-level, so the problem is not simplified. Worse, assuming that this layer contains more than Creation (i.e. requiring more information to describe it), we have merely complicated the problem instead of reducing it to a simpler one (and hence explaining it). Such an assumption is necessary if the Real layer is to meaningfully include God in addition to Creation, except in Pantheistic conceptions. We are thus led to the conclusion that it isn't possible for something that is worthy of being called God - providing ultimate answers - to exist.
In the second case, God creates a Creation that is seperate from his layer's existence. In a sense, then, he merely extended the single Real layer of things. But then there are two options - it is possible that the properties of Creation are determined by the properties of the creating God, in which case they contain no new information and so God does not provide an explanation for why things are the way they are, as noted above. It is also possible that the properties of Creation are not determined by God, in which case they must arise randomly - but then God's qualities (the qualities of the first layer of Real things) does not explain the properties of Creation either. So in either option we are left with a God that does not provide an ultimate answer to why our world is the way it is.
The fundamental error of classical theology is the neglection of the fact that in order to create complex things one needs to have the information to create them in either the original, earlier, things or else have their complexity arise randomly. It isn't possible for a thing to both be the sufficient cause for complexity and be simple; for a simple thing to breed complexity randomness is needed as a source for more information, as the information is simply not there to be brought of the simple thing.
Modern physics embraces both avenues at once. Seen globally, quantum mechanics maintains a unitary evolution within which information is conserved, so that existence is just as simple (or complex) at any time (one naturally tends to assume least complexity, which is the vaccum state - nothingness). From a local prespective, however, observation appears random, creating more apparent information. Thus the properties of our world are explained by random-generation of information from a completely simple genesis, while at the same time recognizing that these are somewhat illusionary as the underlying reality is simple and the information needed to describe our world is merely a result of looking at reality from a tiny, finite, this-world prespective.
A Simple underlying reality cannot explain why our world is the way it is, while a Complex underlying reality does not do any explanation as the complexities are not abated. The resolution of this paradox is randomness, that creates information ex nihilo.
The take-home-message is that the only way to explain why our complex world is the way he is that it is arbitrary - it arises from randomness, there is no way for there to be an ultimate explanation for why things are the way they are.