To let one's guard down and trust someone; to open one's heart and become close/trusting toward another person.
最初は壁があったが、何度も話すうちに彼女に心を許すようになった。
At first there was a barrier, but after talking many times I came to trust her.
初対面の人にすぐ心を許すのは危険だ。
It's dangerous to trust someone you've just met right away.
彼は親友にだけ心を許して、本当の気持ちを話す。
He only opens up to his close friends and tells them his true feelings.