Excessive growth of skin, surrounding (or sometimes covering) a wound; caused by granulation.
Practical translation: some horrible, god-awful stuff that sometimes grows in around/over a wound instead of "normal" flesh. Imagine an ingrown toenail cutting the skin of your right foot's big toe. Normally, trimming the nail, putting an antibiotic ointment on the injured skin and bandaging it up, then treading lightly for a few days is sufficient for it to heal.
But not always.
Now imagine that instead of the wound just healing, a new growth appears. It's a bit more pale than the skin surrounding it. It's incredibly soft, and always just a bit moist.
And it feels like your foot's on fire. That's when you leave it alone. When you touch it, you discover what proud flesh really is: a rat's nest of nerve endings and blood vessels growing almost at random in response to an injury your body is sick of dealing with. Any physical contact with anything, be it by walking, getting dressed, showering, or letting air hit it, causes instant and immediate intense pain.
Proud skin is a demonstration of just how much a tiny, tiny part of a person's body can hurt so much more than just having an arm lopped off.
Treatments for proud skin include patience (good luck), topical and oral antibiotics (sometimes proud skin results from infection), and outpatient surgery to remove the growth.
It was actually less painful for me to be injected with local anesthetic at the base of my right foot big toe, have the proud flesh cut off and the offending toenail partially destroyed (on purpose), and have the whole works wrapped up in thick bandages for three weeks than it was prior to the procedure to walk around on the damned thing. Yes, there was still pain (particularly when the anesthetic wore off and my toe suddenly realized "aaaugh! Someone just cut a bunch of living tissue off me!"), and it was bad, but it was still significantly better than the proud flesh gave me. I cannot properly describe just how painful the stuff is.
You do not want to grow any proud flesh anywhere, ever.