What does it mean?
Bāo zài wǒ shēn shànɡ is a common saying especially in spoken Chinese. It means leave it to me; I will take care of it. Here Bāo means be responsible. “Shēn shànɡ" equals to on the body. The whole phrase shows the meaning: Don't worry, leave it to me.
How to use?
The usage of this phrase is like a sentence. For example:
(1)A: 周六你能帮我搬家吗?
Zhōu liù nǐ nénɡ bānɡ wǒ bān jiā mɑ?
Can you help me move my home on Saturday?
B: 没问题,包在我身上。
Méi wèn tí, bāo zài wǒ shēn shànɡ.
Sure. Leave it to me.
(2)这件事情包在我身上,你别担心。
zhè jiàn shì qínɡ bāo zài wǒ shēn shànɡ, nǐ bié dān xīn.
Leave this matter to me, you don't need to worry.