The answer is quite simple. The sites that get ranked the highest under more competitive keywords are those that have "authority" in Google's eyes under that specific keyword. Authority is achieved through Page Rank and Keyword Authority, both of which are achieved through a process known as "backlinking".
A backlink is basically an internal or external link that points towards a specific web page or web site. If for example you had a site called mysite.com, and another site called banana.com had a link on their site pointing towards yours, you would have a "backlink" from banana.com.
This tutorial shall begin by explaining the basics of Page Rank and Keyword Authority, and how a backlink can affect both. If you already know what Page Rank and Keyword Authority is, it is recommended you skip the following two sections of this tutorial. Remember that Keyword Authority and Keyword Optimization are two different things.
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