The short story is this. By watching what you click on in search results, Google can learn that you favour particular sites. For example, if you often search and click on links from Amazon that appear in Google’s results, over time, Google learns that you really like Amazon. In reaction, it gives Amazon a ranking boost. That means you start seeing more Amazon listings, perhaps for searches where Amazon wasn’t showing up before.
The results are custom tailored for each individual. For example, let’s say someone else prefers Barnes & Nobles. Over time, Google learns that person likes Barnes & Noble. They begin to see even more Barnes & Nobles listings, rather than Amazon ones.
Of course, people will be clicking on a variety of sites, in search results. So it’s not a case of having one favorite that that simply shows up for everything. Indeed, Google’s other ranking factors are also still considered. So that person who likes Amazon? If they’re looking for a plumber, Amazon probably isn’t close to being relevant, so the personalization boost doesn’t help. But in cases where Amazon might have been on the edge? Personalisation may help tip into the first page of results. And personalisation may tip a wide variety of sites into the top results, for a wide variety of queries.