The passage says: "If your enemy is hungry, feed him; if he is thirsty, give him something to drink; for by so doing you will heap burning coals on his head." The idea of "burning coals on his head" doesn't seem to fit with the idea of showing kindness to an enemy, so what does it mean?
NKJV - 20 Therefore "If your enemy is hungry, feed him; If he is thirsty, give him a drink; For in so doing you will heap coals of fire on his head.
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It means he will be embarrassed. The way it is said now is "he has egg on his face." It's just a figure of speech.
Back in bible days fire was very important to daily life. Often if your fire went out people would go to a neighbors to get a bunch of coals (embers) to relight there own fire and of course, like many things, they carried things on their head. So, in being kind,generous, loving to people it helps to relight their own fire of faith