Step 1: At the top of the Weebly editor, click on Pages. Go to the Page that you want to add the meta-tags to. Click on the three dots on the extreme left of the page.
Step 2: Once you're on the page, look for SEO Settings.
Step 3: When you're on SEO Settings, scroll all the way down to the bottom until you see Headers.
Step 4: Add to the Header Code by modifying the meta-tags I listed above.
If you're using Twitter, change the relevant information to link to your Twitter account, and your Twitter website card. You can check that Twitter is reading your meta-tags correctly by using this validator: https://cards-dev.twitter.com/validator.
If you're using Facebook, you'll want to modify the open graph meta-tags. You can check what Facebook is grabbing from your site by using their debugger tool: https://developers.facebook.com/tools/debug/
Note: I used Imgur to generate a shareable link to my open graph images. The ideal size for a Facebook open graph image is 1200 by 630 pixels. The ideal size for a Twitter website card is 800by 418 pixels.