To search your files, type <space> in Alfred's search bar and then you can search for your files! If you want to search for your files automatically, then check the Folders box under General > Default Results.
I'm assuming when you're talking about quitting the app, you're referring to the Alfred preferences window. It runs separately from the search feature meaning that although the window isn't open, Alfred search is still running. To actually quit the Alfred, simply go to your menu bar and click on the Alfred icon. From there you can Quit Alfred (not sure why you'd want