Darwin said that all creatures produced more offspring than were capable of being supported by their environment. So he reasoned that the ones that did manage to survive must have some characteristic (or combination of characteristics) that gave them a competitive advantage in their environment. Whichever creatures within a species that had that characteristic would be more likely to reproduce, and they would then have offspring that also had that characteristic. Through this process, species evolved, even into entirely new species. Darwin used the famous Galapagos finches, which evolved different beak sizes and strengths in response to their environments, as examples of this phenomenon.
The important thing (and the most disturbing to many of his contemporaries) was that he posited that these variations within a species occurred randomly, which seemed to remove any room for a divine plan. It is also significant to note that Darwin wrote before genetics were understood at all (it was seventy years, in fact, before Gregor Mendel's work was popularized and integrated with Darwin's theory.) So he knew nothing of mutations, which do, in fact, occur randomly. In any case, it was through this process of natural selection that evolution, which had been largely accepted by educated people for many years, occurred.