This is a relatively short explanation of atheism from a very pragmatic perspective. It's fairly basic, so do not assume that this will describe any given atheist in their entirety. What it will do is give you some idea of what the basis of many atheists' conclusion is.
Each person reading and comprehending these words exists. To whatever extent, and however they define those terms, we all exist. We exist in reality. This planet, this solar system, this galaxy, this universe. All of this is our reality and we know that it is here. We observe nature. We observe life and death. We observe time and space. We have come to know various aspects of our reality and we can continue to learn new things. This is a simple understanding of observed reality. It supposes nothing, working only with the observed data and building worldview only from what is known.
This is where the atheist and the theist encounter their fundamental and only difference. The theist observes the exact same reality as any other person. The theist then ascribes that reality to the design of an otherworldly being. The planet, solar system, galaxy, universe... all of it exists through and for the sake of the otherworldly being. The atheist does not accept these claims because observed reality bears no evidence in support of them. Whatever connections are drawn between observed reality and the existence of this otherworldly being are circumstantial at best, tenuous on average and complete fabrications at worst. The atheist sees no reason to believe that their reality is the product of the will of an otherworldly being when reality explains itself in much simpler terms with good evidence to back it up.
Hopefully this short description has served it's purpose well. I welcome all comments and criticisms. I do not do much writing of this sort so your input is appreciated