rasmus b. you should have at least one project that you have available to you in Figma. I just dont think you have a project created. When you hover over your [Your name] Team - you should see a plus icon, and you can create a project there. And then once you have a project created, you should be able to move this figma kit to that project. Regardless though, I've heard of people working from drafts folder too - you have editing power to the file.
If you are trying to move it into a specific team that you have joined - you would need permission from that team's admin for editing and likewise for projects.