If it's not immediately obvious why this is a huge component to many atheists' worldview, think about a Christian and his bible. Genesis, which is supposed to be the word of the Lord, describes the making of the world in six days, and when the math in Genesis is added up, the Earth should be about 6,000 years old. It also says all living beings, including humans, we're made by god during this period.
But, because of science, we know none of this is true. The Earth is 4.6 billion years old, and evolution proves all living things came from a single-called organism and eventually LUCA, which stands for Last Universal Common Ancestor. Life has been on this planet for millions of years. So, imagine being a Christian with a curious mind, one who doesn't exactly buy the Genesis myth, and you learn we're a product of natural selection and genetic speciation. All of a sudden, your views on god and religion start to tear at the seams.
Being an atheist means having a rational mind. It means you have to use critical thinking when you look at your life and when you consider your future ... and past. Non-believers look at the planet and all of its living things and think it makes much more sense that we all came from one spark of life and evolved over billions of years through gradual change, rather than some inexplicable deity snapping his fingers and making dinosaurs and humans roam the Earth together 6,000 years ago.
Here's a fun way to screw with a Young Earth Creationist. Ask them if they believe we can see some stars that are 50,000 light years away, which is true and there are plenty that are millions of light years away. When they say, sure, I think there are stars we can see that are that far away, ask them how it is that we are seeing them? If it's 50,000 light years away, that means it has taken the light of that star 50,000 years to travel close enough for us to see it. If they stare at you like a deer frozen in headlights (or in this case starlight), that is when you tell them they just admitted the universe is at least 50,000 years old, totally refuting Genesis. And then you can clue them in that the universe is at least 13.7 billion years old, which is scientific fact.
Now, back to natural selection. It's exactly what it sounds like: Nature selects those best suited to survive its surroundings. What does that mean? Imagine two species of birds living on an island, and the nectar of life is nestled deep inside a long fluted flower. Nature is offering sustenance to whichever bird can get to it. So, which species survives, the birds with long slender beaks or ones with short fat beaks? The long slender beaks will reach the nectar, remain strong and will mate with other long-beaked birds, while the short beaks will eventually die off. It's a harsh view of life, but that's the way we evolved.
Sometimes a genetic mutation will give one species an advantage over another. There is a species of sloth that has one finger that is freakishly large and longer than the rest. It uses this finger to reach deep into the small holes of tree bark to fish out bugs. This mutation made it much more adapted to live in the trees, giving it a distinct advantage over other sloths. To some believers, it may look like their deity made these species and put them in the specific places in the world where they could survive, but does that really make any sense? If you think that, then how come 99.9% of all species that have ever lived are now extinct? Free will? Now that's funny.
But, survival of the fittest doesn't mean survival of the strongest, as shown by my aforementioned bird example. The long-beaked birds weren't stronger than the short-beaks, they just fit in better with the surroundings where they lived. If the short beaks were on an island where they needed to use the leverage of their stubby beaks to break open nuts then maybe they would be the survivors while the long-beaks would have starved. And in the case of the long-fingered sloth, it literally means it "fit" better than its competition.
One final thought on survival of the fittest not being a synonym for strongest: When an asteroid crashed into the Yucatan Peninsula 65 million years ago, it created an ice age that wiped out all of the dinosaur giants. But, tiny marsupials went deep below the ice and lived underground to survive, eventually becoming upright walking bi-peds (that means us to be more to the point). And certainly a T-Rex is stronger than a marsupial, right?