to sink deeply into one's mind; to come home to; to go to one's heart
to pierce one's body (e.g. of wind, cold, etc.)
厳しい現実が身に沁みる瞬間だった。
It was a moment when harsh reality sank deeply into me.
一人暮らしをして親の苦労が身に沁みた。
Living alone made me truly feel how hard my parents had worked.
先生の言葉が身に沁みました。
The teacher's words sank deeply into me.