“may not take her again.” Divorcing a wife was easy in the biblical world, as we see from Deuteronomy 24:1-3, which could lead to men divorcing their wife in the heat of passion, but then wanting to reconcile later. This law, that if he divorced his wife and she married again, that she could not later go back to her first husband, would have given husbands a reason not to divorce a wife in the heat of passion. Ahasuerus divorced his wife Vashti when he was drunk and angry, and seems to have regretted it later (Esther 2:1). Thus this law may have helped save many marriages that would have otherwise ended in divorce.
“after she has been made unclean.” Sexual intercourse made a person ritually unclean, so the fact that the woman had married another man made her unclean to her former husband. However, interestingly, she could marry a third man whom she had never been with before.