How do the finches of The Galapagos observed by Charles Darwin prove the theory of evolution?

Birds in the Galapagos had widely varying beak shapes and coloration, reflecting their varied diets and resembled a variety of birds seen in various places in the world.  Upon close anatomic study, they all had anatomic features unique to finches.  It was apparent that a few finches arrived from South America and each subpopulation diversified based on the features most favorable to utilizing the various food sources.

As Darwin observed, there is a huge range of variation within the single species of the finches. Especially when you look at the beaks - which is one of the things Darwin did.
The beaks range from being tiny, able to pick up very small seeds, to huge, to eat fruit and nuts or insects, to very long and thing, able to pick grubs from deep holes.
Each bird has found a niche and adapted to be able to get what nutrition it needs, with as little as competition as possible.
The birds with big beaks can't pick up the tiny seeds with the ease of the ones with the small beaks, in the same way that the birds with tiny beaks can't get through the tough shells of nuts or skins of fruit.
Most of the finches could only be found in this one area of the world, so it follow reasoning that they had adapted/evolved.
Then you start to go into Natural Selection.

The Galapagos islands offered a great view of natural selection at play. The different finches from the separate areas each had characteristics suited for their environment. Charles Darwin noted these characteristics and the crucial role that the environment played in those characteristics. This is the basic gist of his idea of natural selection. Natural selection states that some selective pressure from the environment (the non-living environmental factors like temperature, terrain, pH, salinity, and such as well as living factors such as available plants and other animals and bacteria). This selective pressure leads to the survival of certain individuals within a population that share an advantageous characteristic. The idea of natural selection came to Darwin from his visit to the Galapagos islands.

Your question is a tiny bit off, though. The finches provided the framework for Darwin's ideas, but the proof of evolution (at the very least, the very solid observations in concurrence with the theory of evolution) include genetics and biochemistry as well as biology in general. Bacteria can show natural selection quite easily. Biological observations such as the morphology of animals and plants provide a great boost to the theory of evolution. Most importantly, genetics has provided a great way to provide the backing of evolution.

Lastly, evolution stems from natural selection. You must understand natural selection itself to understand evolution. I highly suggest reading a bit more about genetics and books such as "The Selfish Gene", "The Blind Watchmaker", or Darwin's own "On the Origin of Species".  I should note that the first two books are by Richard Dawkins, a world renowned biologist but also well-known for his books on atheism. If the atheist bit bothers you, I would suggest other authors that discuss evolution without mentioning anything about the more sensitive subject of religion (though I've read both of the books and did not find them overtly offensive).

Here are some links:

http://evolution.berkeley.edu/evolibrary/article/0_0_0/lines_01
http://www.terindell.com/asylum/jason/darwin.html#finches
http://www.gct.org/darwinfact.html
http://en.wikipedia.org/wiki/The_Selfish_Gene

I highly recommend reading some books on this subject. It happens to be my favourite subject in science.