The short hand version of Evolution
In nature, we observe the following things:
1.) Animals reproduce, but they do not reproduce exact copies. children look like their parents, but not exactly. (there is variation )
2.) These differences between generations tend to be small, but also unpredictable in the near term. So a child is taller or has an extra finger, but they’re not taller or extra-fingered because their parents needed to reach high things or play extra piano keys. (so the variation is random, rather than being a direct response to the environment)
3.) Animals often have more kids than the environment can support and animals that are BEST SUITED to the environment tend to survive and reproduce. So if there is a drought, for instance, and there is not enough water, offspring that need less water—or that are slightly smaller and so can get in faster to get more water—will survive and reproduce. (there is a process of natural selection which preserves some changes between generations in a non-random way)
As a result, over time, the proportion of traits (what we would now refer to as the frequency of genes in a population) will change, in keeping with natural selection. This is evolution.
The video below is also a great explanation, if you can ignore some gratuitous shots at the beginning, the explanation is very clear:
In addition we highly recommend you check out this book Why Evolution Is True by Jerry Coyne.