The correct saying is "bruise like a peach".
"I bruise like a peach" means that the skin of the person who said that is very tender and bruises with the slightest bit of pressure.
Given their rather fragile nature, it is not unexpected to find the idea of injury associated with fruit. Strawberry, for instance, is used rather evocatively in the US to mean ‘a sore or bruise, especially one caused by friction with the ground’.
The phrase to bruise like a peach also gives rise to the more pleasant peaches and cream applied to someone with a lovely and healthy complexion. If something is described as a peach, then it is usually something quite exquisite or a particularly fine example of its type.
See other inventive uses of food in English at Oxford Dictionaries.