The Markup Validator is a free service by W3C that helps check the validity of Web documents. Most Web documents are written using markup languages, such as HTML or XHTML. These languages are defined by technical specifications, which usually include a machine-readable formal grammar (and vocabulary). The act of checking a document against these constraints is called validation, and this is what the Markup Validator does.
W3C stands for the World Wide Web Consortium who are the recognized standards organization for the world wide web. These standards are then used to help guide web developers and browsers to develop code that lives up to certain standards
In other popular terms, with W3C you can check if your code (HTML, CSS for example) is written correct and the spiders can "travel" into it without any problems. It can affect you SEO related, because the errors occurred may intrerupt the bot, or they can't reach the right structure of your website.
It checks your website (your page in this situation!) for errors, mistakes in your HTML code, tags, etc.