I just wanted to add to the "don't feel dumb" crowd because the kind of problem you were having is actually quite common (not knowing how to deal with DMG files). If any of us were standing there with you or screensharing or something visual in nature, any of us could have both shown you how to fix the issue and explained what's going on much better in like 30 seconds.
It's only the nature of a text-based medium like this -- where terminology and experience play a huge role in being able to articulate either the issue or the solution play a huge role -- that it seemed so complicated.
I tell my students that a DMG file is the digital equivalent of a package from Amazon (because everyone's gotten one of those at some point). It's a box. If you open the box, you see a gift-wrapped package. That's the mounted disk image. If you open THAT, you see the actual thing you downloaded (the application or the installer).
As with their real-world counterparts, once the installer has been run or the application has been moved to the Applications folder, you clean up the gift-wrapping and the box and put them in the trash.
