To encourage or push someone (figuratively) to make a decision or take action; can also mean to physically give someone a shove but is commonly used figuratively.
友達に背中を押してもらって留学を決めた。
My friend encouraged me, and I decided to study abroad.
上司の一言が彼の背中を押した。
A single comment from his boss gave him the push he needed.
試験で緊張していたので、彼女はそっと彼の背中を押した。
He was nervous about the exam, so she gently pushed his back.
親の励ましが子どもの背中を押すことがある。
A parent's encouragement can sometimes give a child the push they need.