It works like this: first, you have to save whatever you are working on if you've made any changes. The reason you have to save is that OS X assumes that when you drag and drop, you'll want all the latest information and updates included in your data. Saving makes that happen.
Saving also makes the icon go from a light grey to a dark grey; you can always tell if you've made unsaved changes in these applications because of the icon's color. Go ahead, give it a try in TextEdit. Type some text and then save the file to disk. The icon appears for the first time (in dark grey) once you save your untitled file, and then if you start typing, the icon goes light grey until you save again.
After saving, you can drag the proxy icon to another application and drop it onto either the application icon or into an open application window, or a Finder window. To perform the drag, click on the proxy icon and hold the mouse button down for a second until the icon darkens. Once it goes dark, you can drag it.
The results of dragging the proxy icon are going to vary by source and target application. In most cases the effect should be the same as if you had dragged the Finder icon for the document. For example, if you drag a TextEdit document onto Mail or Eudora, it creates a new letter with your document as an attachment. If you drag onto a Terminal window or into another TextEdit document, you add a bit of text that shows the path to your document, e.g. /Users/ericasadun/Documents/mytesttext.txt. Drag it to Safari and it opens and displays the document in a new window. You can also drag pictures, videos, audio and other file types.
Keep your eyes open and look for the proxy icon in your day-to-day applications.