It's a dropbox-style cloud, which means it's a folder on your local computer and it syncs to the cloud in the background.
Their Cloud files have useful understanding of the Adobe filetypes and can let you preview on the web, do annotations, turn on and off layers, and download in different formats. For example, I could put an Illustrator file up there and you could toggle layers then download as a png, jpg, AI or a PDF. Those are pretty decent, but the cloud files are definitely not the strong point of Adobe CC right now. Their mobile apps could sync to CC, but they've discontinued most of those except Ideas.
The strong points right now for the cloud piece of CC are the TypeKit account, the 5 basic Business Catalyst sites, and the Adobe Edge & Muse apps (which are only licensed through CC).