I opened my computer to clean the dust out. I put it back in the case seemingly with no problem and now the drive slots will not open. What did I do wrong? Thanks

I'm kind of an amateur but even I would not have taken the actual case off to clean dirt out. The only thing I could tell you to do is turn it the entire system off and turn back on and see if that helps. Maybe the system needs to reset itself. Also, be sure the ribbon connections for the CD and DVD drives are properly seated. Ouch!

Yeah this happened to me before and its cause I wasn't careful enough and the ribbon connections came out. Its easy enough to fix.

It's OK to open the case to clean out all that dust , its better that the PC overheating.(just be careful about static discharge inside the case , don't use your home vacuum or touch the mobo).
The CD drawer will open as long as it is getting power , the 4 wire Molex Molex - Wikipedia, the free encyclopedia , provides the power .
The IDE ribbon cable needs to be connected to allow data transfer.
If all else fails and you need to get a cd out of the drive there is a little hole at the front of the drive where you can insert an unbent paperclip that opens the drawer.