Why it is Important to Keep Your Site Standards Compliant

By validating your pages you make sure the structure of the document is acceptable to standards-compliant browsers.
When you deviate from the standard, you don't know how the browser is going to react.
Saying that "I don't know if my site is standards compliant or not, but it looks good in all browsers." (or just in one browser and one platform on which it was tested) is not a good idea.
The browsers may not display this page correctly in future versions, or a new browser may do things differently.
(For instance, when MSIE 5.0 came, it was more standards-compliant than MSIE 4.x and as a result many sites that worked with IE before became broken. This happened in the release of MSIE 6.0, and in Windows XP Service Pack 2, as well.)
The browser has a tough job as it is, so you shouldn't make it tougher by sending it mal-formed input.

Caveats

Standards-compliance is not a panacea.
Even if all the output from the site is standards-compliant it doesn't mean it will display correctly.
This is either because of bugs in the browsers, or because you haven't done the right thing.
So, it is important to test the pages in as many browsers as possible, in addition to validating them.