A Destructive Analyzer is a machine that deconstructs components to reverse-engineer their technologies. Deconstruction itself is easy. Simply stick items into the destructive analyzer and turn it on. It is possible to eject items from the deconstructive analyzer if, say, security notices their missing laser guns. As always, experiment! Some technology on the station is just so basic that the analyzer won't even bother with it, but things like bruise packs, flashbangs, and some of the odds and ends in your very lab will all work.
For related devices see:
List of Destructible Items
You'll only advance one level at a time per item, and only if the tech level is equal or greater than your current research level. Higher research levels will allow additional items to be created using the Protolathe and Circuit Imprinter.