'Putting away' is mostly translated as divorce. My understanding of this is that, in OT times men could divorce their wives with relative impunity, for various reasons. But Jesus said that it should not be so. Matthew 19:9 & 5:32 suggests that divorcing an adulterous wife is acceptable; but other than this, it seems that Jesus is saying that marriage should be taken more seriously - and that attempting to divorce for other reasons is breaking a solemn vow (acting treacherously).
Discussion
Suppose a man has dealt treacherously by divorcing his wife and marrying another, thus committing adultery.
Why is it adultery for another man to marry the wife he put away?
Luke 16:18: "Whosoever putteth away his wife, and marrieth another, committeth adultery: and whosoever marrieth her that is put away from her husband committeth adultery."
Because in God's eyes the original marriage still stands and hence the wife is not free to marry another?