Darwin provided what I believe to be an essentially correct interpretation of how the finches came to be the way they are. A few individuals were probably blown to the islands from the South American mainland, and today's finches are descendants of those pioneers. However, while Darwin saw the finches as an example of evolution, we can now recognize them merely as the result of recombination within a single created kind. The pioneer finches brought with them enough genetic variability to be sorted out into the varieties we see today. Now to consider the third source of variation, mutation.
Mutations are mistakes in the genetic copying process.
Each living cell has intricate molecular machinery designed for accurately copying DNA, the genetic molecule. But as in other copying processes mistakes do occur, although not very often. The cell has machinery for correcting these mistakes, but some mutations still slip through. Some have no effect at all, or produce so small a change that they have no appreciable effect on the creature. But many mutations have a significant effect on their owners.