To engrave something firmly in one's heart; to take a lesson or warning deeply to heart and not forget it.
師の言葉を肝に銘ずるように常に心がけている。
I always try to keep my teacher's words engraved in my heart.
今回の失敗を肝に銘ずるべきだ。二度と同じ過ちは繰り返すな。
You should take this failure to heart. Don't repeat the same mistake again.
彼女は『危険は常にある』という言葉を肝に銘ずると語った。
She said she keeps the words 'danger is always present' in mind.