Literally 'wrap it on my body' — the image is of strapping responsibility onto yourself. The strongest 'leave it to me' in common Chinese: full guarantee, chest-thumping confidence. Slightly old-school / heroic flavor, often used half-jokingly among friends.
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Picture literally wrapping (包) the problem onto your body (身上) — if it falls, it's your fault. That's the commitment level 包在我身上 promises.
