That is, if the mechanism designer wants to implement some outcome or property, they can restrict their search to mechanisms in which agents are willing to reveal their private information to the mechanism designer that has that outcome or property. If no such direct and truthful mechanism exists, no mechanism can implement this outcome/property by contraposition.
If a mechanism designer wants to create a system where certain outcomes occur, they must find mechanisms where agents are willing to reveal their private information to the designer who has that desired outcome. If no such mechanism exists, then no mechanism can achieve that outcome.