to soak into; to sink into; to seep into; to permeate
to sink into (one's mind); to be (deeply) ingrained; to be instilled; to be embedded; to be impressed
雨が服に染み込んでしまった。
The rain soaked into my clothes.
教えが心に染み込んだ。
The teaching sank into my heart.
音楽が心に優しく染み込む。
The music gently seeps into the heart.