Verifying your site with Google Webmaster Tools gives you access to reports about your site's visibility and ensures that your site is indexed by Google on a faster / more regular basis. It isn't necessary, but it's a smart thing to do.
Follow the link above and then log in (if you already have a Google account of some kind) or create a Google account (if you don't already have one). Any Google account will allow you to log in. Use the Add a Site button to add your site to the Webmaster account. You can add multiple sites to one account.
This will open the Verify Ownership page. You have to verify yourself as the owner as Google doesn't want just anyone to be able to represent your site.
Select the option to "Add a meta tag to your site's home page" and then copy the provided tag.
Don't click the Verify button until you've added this code to your site:
Go within the editor of your site to Settings > Search Engine Optimization and paste the full code into the Header box in this section. Then be sure to Save and Publish your site. If there's already code in the header, put this code above or below it.
When you've done this, go back to Google Webmaster Tools and click the Verify button referenced a couple of paragraphs above. You should leave this code in place even after the site has been verified.
You'll now be on the Dashboard page. From here, you submit your site's sitemap. Sitemaps are automatically created for sites on Weebly at http://example.weebly.com/sitemap.xml or http://www.yourdomain.com/sitemap.xml in the case that you're published to your own domain name. Either way, you just need to enter sitemap.xml into the box Google provides you when you submit.
All done! It may take up to 48 hours to get verified, but then you'll be all set.
Google Webmaster Tools has a lot of cool features to play around with after you're verified. These features can help you see how well your site is ranked You never have to use them (verification is the only significantly important thing to do here), but they're there if you want to check them out.