Your new web host MIGHT have a "free offer" to extend your domain name for a year, if you transfer your domain registration to them. You do NOT have to transfer your domain registration to them.
I'd read the fine print carefully. We see a lot of tales of woe here from people who didn't read the fine print. Free isn't necessarily cheap.
In particular, make always make sure that YOU are the registrant of your domains.
Domain registration, web hosting, email hosting, name servers are completly seperate services, and anybody "bundling" these without making it clear that they are completely seperate services that you can comparison shop and go anywhere for is trying to sucker you.