Basically, you don't understand how it could happen, therefore conclude it didn't.
But it's not that complicated.
Observe the following, which is the standard dogma of genetics:
DNA<-->DNA-->RNA-->protein-->trait
Mutations (which provide an innate variability to all life, and are largely a result of copying errors of DNA) happen at the level of DNA.
Now, "mutation" doesn't mean "makes a monster." Mutation is some change to the DNA - the addition, deletion, or changing of nucleotides in DNA, which are the basic building blocks of nucleic acids (DNA and RNA).
Selection will act at the level of "trait" if the trait in question has a basis in DNA. Selection just describes a process of induced differential reproductive success - some organisms with a particular genetic makeup tend to pass on that genetic material better than other organisms, based on particular criteria.
What is that criteria? Different organisms live in different environments. Within a given environment, any particular organism has a "job" in respect to its overall place in a community. That "job" (called an ecological niche) describes what they're eating, where they live, how they're interacting with other organisms and each other etc. New jobs come and go, old jobs are lost, changed hands, etc. There's all sorts of stuff that can cause this, but these are the background factors and the basis by which nature "selects" organisms most suited towards a particular role.
So, natural selection is steered by the above - if a trait in a particular organism is advantageous towards a particular niche (job) in nature, that organism is more likely to reproduce and pass on its genetic material to the subsequent generation.
Over time, this process shapes the "gene pool" that represents a given population. From one generation to the next, the change you're going to see will be fairly subtle. So it's not like one organism is born with a radically different genetic code, he's super organism, and then passes on his super genetic material to the next generation. I have to stress that the change from one generation to the next is almost always subtle - it's when that change cumulates over deep time that you notice "big" changes.
The overall size of the gene pool during this process may increase, decrease, or stay relatively the same. But the overall pool is "changing," thus it's fair to define evolution as "a change in allele (versions of a gene) frequency in some population over time." I repeat - this "change over time" of that gene pool IS evolution, and it's adaptive because nature is steering that process. It's ensuring that genetic material which is advantageous for individual organisms is propagated at a higher frequency than less advantageous genetic material.
This combined with the description offered to Carico above is evolution in a nutshell. It might be hard to see how all these changes can add up, but that is EXACTLY what happened. All known evidence confirms this reality, and while it might at first seem counter-intuitive, mindful study of the subject will lead you to this conclusion.
Edit: if you're further confused how a population undergoing all this change manages to "make it," consider the fact that the vast majority of creatures that ever lived are extinct. So most "don't make it." The organisms you see in the world around you are the ones (or descendents of those) that did "make it."