Hello! This is an interesting question. I think the original form of the phrase is 'turn something to someone's advantage' which means using or handling something in a way that a person can benefit from it. In this case, that particular 'something' is an 'innocent occasion', like the selection procedure of the champions for the Triwizard Tournament. In my interpretation, what Moody meant to say is that there are people in this world who would use every occasion, even the simplest and most innocent ones, to gain something from them for their own benefit. Moody may have been paranoid, but we do know that whoever altered the goblet of fire, did so in order to capture Harry and use him for their own dark purposes. By saying this, Moody accused Karkaroff for perpetrating this act, obviously because of the latter's past association with the Death Eaters. I hope this explanation will help you to get a better understanding of this phrase.