There doesn't seem to be a way to show the desktop straight from within the full screen app, but (depending on what methods you use for various kb/mouse shortcuts etc) it can still be done without exiting full screen (although someone may show that doing so would actually be quicker, perhaps, than my method below).
Let's say I have Safari full screen, and I want to view a screenshot by opening it in a new tab (not a particularly useful scenario, but involves dragging a Desktop icon into a full screen app as per your question). Here how I would do it, noting that there are probably loads of other methods depending on your preferred input methods. The bold bits are the actual actions I choose, the rest is commentary on why:
- When I switch Safari full screen, it creates a new space to the right of the one I am in.
- From within the full screen app, my desktop gesture of choice (5 finger trackpad spread) doesn't work (neither does F11, but then on my keyboard it's the volume down key, and I don't have "use all F1, F2, etc keys as standard function keys" preference option set, and the FN key is not located particularly well on a full size desktop keyboard, unlike the laptop ones)
- So I 4 finger swipe to the right to show the previous screen before I went full screen (you may do ⌃Left Arrow)
- Then I 5 finger spread on my trackpad to show the desktop (You may use F11 if it's not been commandeered by the volume down key or FNF11 or whatever the key happens to be depending on the age of your keyboard)
- Then I hardware click with my thumb and hold the icon I wish to drag with my trackpad
- Then I 3 finger swipe left (which exists the Desktop view and then swiped across a desktop in one movement)
- If you are not using a trackpad you can hit F3 once to exit desktop mode, and F3 again to show Mission Control (your mapping and or keyboard layout may differ), then just hover over the required full screen desktop at the top until it flashes and switches you to it in a second or 2)
- And I am left with the file still help under drag conditions to drop wherever I want within the full screen app
This method uses a variety of inputs including gestures and keyboard shortcuts that may not be available to you depending on your hardware. For example, using the trackpad allows me to perform simultaneous gestures like swiping desktops whilst in the middle of a drag operation that would not be possible with a mouse (maybe with a magic mouse, I don't own one). It's a case of knowing that gestures are actually pretty powerful when you understand that the hardware uses the multitouch to let you do a lots more than you might be used to.
This seems a whole load of hassle, but as actually quite quick once you get into it. As to why you do it this way, I think you need to contemplate the nature of full screen of Mac OS X, and this means appreciating that when you go full screen you lose the whole desktop metaphor, and this includes the desktop in addition to resizable window edges, so you need to quickly nip into a windowed desktop to be able to see behind those windows to the desktop, if you see what I mean.