Your Basecamp files appear in Finder as "cloud" files, just like in Dropbox or iCloud Drive. They're there, you see them and search for them, but they don't take up space on your disk until you open them. This way you can have huge projects set up without filling up your Mac.
Folders: projects and their Docs & Files
Each project you activated is a folder in the Finder sidebar. Inside are the folders and files just as you have them in Basecamp's Docs & Files, with the same structure.
The cloud icon
A file you haven't downloaded yet shows a cloud icon. When you open it, Companion downloads it at that moment and the icon disappears.
Stays up to date on its own
Companion checks Basecamp every few minutes and also when you enter a folder, so what you see reflects what's in Basecamp. If someone on your team uploads or changes something, it appears in your Finder without you doing anything.
Note: The first time you enter a large project, give it a few seconds to load all its content.