To use a domain name you purchased with Google, you'll first want to change your site address in the Settings tab in the editor. Save your changes and copy the IP address you're shown. Then login to your Google Domains account and go to the My Domains section.
Click on the DNS button to the upper right of your domain and scroll down to the Custom Resource Records. You may have some records already set up that you can edit, but we'll take look at how to add them from scratch.
You'll need to create two A records for your domain to make sure it's properly pointed to Weebly. There are four fields to fill out, but you only need to complete three of them. For the first A Record, enter the @ symbol into the Name field, select A from the Type dropdown menu, and then enter the IP address you copied earlier into the Data field. You can leave the TTL field the way it is. When you're done, click the Add button to create the record.
For the second A record, enter www into the Name field, select A from the type dropdown menu, and paste in that same IP address you used on the last record. You can leave the TTL field as-is again. Click add when you're done, and you should see both of your new records below.
That's it! The only other thing to do is publish your website to the new address when the changes have gone through. In some cases it can take Google up to 24 hours to update their records with your changes, though generally it should only take a few.
What if you already have other records besides the two A records? That's normally not a problem, and you shouldn't need to do anything with the other records. The only time you will want to remove an existing record is if your domain has an AAAA, or quad-A record. You'll need to delete this if you do have one, as these records aren't compatible with Weebly and it won't be possible to set up SSL for your domain with this record in place.
You can check to see if you have an AAAA record in the Custom Resource Records section. It might look something like the one in the image below. All you need to do is click the delete button and confirm that you want to remove the record.