If Weebly is trying to charge you a monthly fee just for your domain name then I would look elsewhere to purchase your domain name. A domain name typically costs anywhere from $9-$14 per year depending on who you register it through.
Weebly's monthly fee may also include the web hosting cost to host your domain. Keep in mind that a domain name and web hosting are two completely different things.
A domain is the URL you or anyone else types into their browser (Firefox, Chrome, IE, Safari, etc). You need to register a domain name so that you own a piece of "real estate" so-to-speak on the internet for people to access.
In the most simplest terms, web hosting for your domain is needed so that your site (URL) can be accessed online. There are costs associated with web hosting (server costs, support costs, and other operational costs) that many people aren't aware of. This is why pricing varies so much from providers because costs may differ on the back-end (server and other infrastructure costs).
If you're still looking to start your blog feel free to reach out to me and I'd be more than happy to help!