I was browsing the internet and noticed, YouTube, for example, contains a URL like this to denote a video page: http://www.youtube.com/watch?v=gwS1tGLB0vc
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File extensions are not used because of the idea that URIs (and therefore URLs) should be independent of implementation - if you want to access George W. Bush's addresses, you should be able to go to http://www.whitehouse.gov/presidents/georgewbush/addresses (for example). Whether the White House's servers are using PHP or Python or Perl doesn't matter to the end-user, so they shouldn't see it. The end-user doesn't care how the page was generated, because all web languages output the same HTML, CSS, and the like, and they're just viewing the page in their web browser.
Most web frameworks build this functionality in by default, precisely for this reason, and it can be accomplished regardless with URL rewriting in most webservers. This ideal is codified in the W3C Style Guide, which is undoubtedly a big proponent in this idea being so widely accepted. It's outlined in their guide, "Cool URIs Don't Change", which should clear things up if you still don't quite understand the reasoning here. That document is the go-to statement on the issue, and the de facto standard for frameworks.
It is worth noting that usually files that end up being downloaded (and sometimes data files used in AJAX) will still have their file extensions intact - http://example.com/song.mp3 or http://example.com/whitepaper.pdf - because they are intended to be saved to the end-user's computer, where file extensions matter. The extensions are not included for pages that are simply displayed - which is most pages.