This website verification process is similar to the options available when verifying through Google Webmaster Tools or Bing Webmasters, Pinterest offers you a file to download and once it has been uploaded to the root domain of your website, you can click to verify the file and therefore your website.
This is what the top of a profile looks with a verified profile:
Word to the wise: you cannot verify the same website on 2 different profiles (a lesson I learned the hard way). Verifying your website can make your URL stand out (as it now shows the entire domain instead of the website icon) and it also makes your profile more legitimate.
How to you make your website more legitimate? Let us know in the comments.