Ignore what you know about "natural selection" and "fitness" and whatever. If you don't really know about that, don't worry about it. We'll get there by a different and more abstract path.
You tend to see things that are common, at least in the set of things you encounter. Things are common to the extent that they are good at being common. There are two aspects that determine how good something is at being common:
Among things that are complex and can reproduce — basically, living organisms — coming into existence mostly depends on reproduction. For living organisms, staying in existence is called "survival".
Thus, the organisms you will see most often will be those that are best at reproducing and surviving.
Much of how an organism functions depends on some pattern intrinsic to the organism. (This is understood as its genes, but the explanation here should be acceptable without depending on deep knowledge of genetics.) Sometimes, this pattern — those parts of an organism's mechanisms which are copied in reproduction — changes, changing the organism's function accordingly. In our world, those changes are random. If the change affects the organism's ability to survive and reproduce, that organism will become more or less common, and you will see it more or less often.
Changes in those intrinsic patterns (genes) are preserved over generations (unless overwritten by other random changes). Thus, the set of most common organisms changes over time in a way that (on average) reflects what functions contributed to effective survival and reproduction in the preceding time.
This is the only mechanism of evolution. Some corollaries of it:
This is the basis of evolution. It's more complicated than this in practice, but the rest is commentary.