You should pay attention to the word “pure.” What does “pure” mean? It means being pure and clean, free from all defilement. Not having the slightest defilement is the kind of purity in which not even a single thought arises.
Not killing includes purifying our thoughts. If we uphold precepts on the surface and claim that we don’t kill, but we complain about others constantly or get angry with people, we are still violating the precept against killing.
Not stealing includes mental states. We don’t have to steal physically. When we envy other’s wealth or talents, or become jealous of people, we transgress the precept against stealing.
Not committing sexual misconduct includes not having improper thoughts about the opposite sex. If we indulge in improper thoughts about the opposite sex, then we are being impure and breaking the precept.
Holding the precepts purely means leading a clean life without any defiled thoughts. That is to hold the precepts purely.