1. How Darwin came up with the theory of natural selection:

Darwin's theory of natural selection was largely based on observational data that he collected during his voyage around the world in the ship called the HMS Beagle. During this voyage, Darwin observed that different versions of similar creatures inhabited continents on opposite sides of the world. Darwin concluded that these organisms stemmed from the same ancestor. He reasoned that the species had variations. Members of the species that had advantageous characteristics within a certain location survived the external elements. Those with less advantageous characteristics did not survive.  

In other locations, such as the isolated Galapagos Islands, there were species that were not found in any other part of the world. He reasoned this was because of the unique, isolated environments of the islands. 

2.  Examples Darwin used to support his idea:

The species found within the Galapagos islands are often used to support Darwin's ideas. Finches with slightly different beaks live on each island. Darwin concluded that the beak shape was related to the food that was available on each island. The finches that could eat the food on that island would survive. The finches that had beak shapes that prohibited them from accessing the food type on that island would eventually starve to death. 

3. How he disproved the work of other scientists:

Lamark was one scientist whom Darwin countered. Lamark is accredited with the Theory of Inheritance of Acquired Characteristics. Lamark claimed that animals consciously changed their physical characteristics in order to better survive in their environments. For example, giraffes have long necks because they purposefully stretched their necks out over time in order to access leaves that no other species could reach. Each generation stretched their necks out a little more and passed this trait onto their offspring until the necks reached their correct length. This theory did not match up with Darwin's, which postulated that an organism's desires did not impact their species' evolution.

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