The bottom line is that to a person whose worldview of reality ("all that there is") is that which occurs in nature - the arguments are equally asserted to rationalize the metaphysically naturalist (or atheist) worldview. For instance, that God is an unnecessary hypothesis - or that physical laws and constants had to be the way they were for physicists to identify them - or that someday a physical explanation will be given for everything.
Nowadays when people refer to Occam's Razor, they often express it more generally, for example as "Take the simplest solution".
The relevance to atheism is that we can look at two possible explanations for what we see around us:
There is an incredibly intricate and complex universe out there, and there is also a God who created the universe. Clearly this God must be of non-zero complexity.
No, it isn't. Occam's razor is an operational principle of science and has nothing to do with metaphysical naturalism. (Unsurprising, considering William of Occam was a theologian and a monk). The Anthropic Principle, in whatever variant, is an attempt to deduce something about the Universe from the fact that we exist. One is not the other. js1138 said he saw no need to invoke 'information theory' to explain evolution. That would simply be an application of Occam's razor. It implies nothing about the existence of a god or the tendency of that god to intervene in the current evolution of the universe.
As for solipsism, it can adequately be refuted without invoking Occam.
Existance causes itself? Mastrubatory or tautologic. Still a deism. And flawed too, even accorded to Occam.
So what makes G-d a simpler theory? Only that we are as we are. "We are as we are" is not the same as "It all is".