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.
The passage explains that if a mechanism designer wants to achieve a particular outcome or property, they must find a mechanism where agents are willing to reveal their private information to the designer who possesses that outcome or property. If no such mechanism exists, then it is impossible for the desired outcome to be achieved through any mechanism.