Your just getting a new address
When you switch to Convert, we simply redirect your domain name to your new Convert site. Nothing actually happens to your old site.
Here's a quick illustration of what happens:
Imagine you just bought a beautiful new home, it's everything you ever wanted. So when your friends call you to visit, you give them the new directions how to get there...you wouldn't want them showing up to your old house of course. Providing you don't sell your old house, you could still move back in anytime, your old house stays where it always was.
It's the same with your new Convert website. We simply change the directions to your website, leaving your old website right where it is.
It's likely you pay for hosting for your old website. Of course with Convert, you never need to pay for hosting again. So if you choose, you can cancel the hosting on your old website. If you do that and want to keep a copy of the old site, make sure to download it from your website host before cancelling your account with them!