Why does the surrounding skin peel off after an injury heal?
By the surrounding skin, I think you mean directly around the site of the injury.
If the skin has been cut, grazed, burnt, or has become infected - with acne for instance, that area will go through four recognisable stages of healing, before revealing new skin.
The first stage, immediately after the injury has occurred (the hemostasis phase) is when the blood in the site clots. The blood platelets form a fibrin mesh, which keeps the clot stable and strong.
The inflammation phase then starts and continues for about two days, longer if the injury is severe. This is when the body cleans up the site, killing bacteria and pushing out surface debris. Macrophage cells flood in and secrete proteins that stimulate immune cells to do their bit and start repairing the tissue. You will see some redness and swelling during this phase, the extra fluid stretches the skin slightly, but this offers some surface protection to the area.
When this is done, the skin can start to replace what has been removed - through the trauma - using connective tissue, and if the injury is severe, new blood vessels. The edges begin to contract and new epithelial cells are produced to cover the damaged site.
In the final stage, new cells begin to grow, the collagen and elastin get going and make the skin strong and flexible. There can often be signs of stretching or puckering as these new cells get comfortable. This can take a couple of weeks, months if the site is badly injured.
The initial swelling will markedly reduce as the extra fluid subsides. As this happens, the skin in that area begins to shrink and dry, allowing it to slough off. It has been stretched, drenched and then contracted, so as it dries out, it is thin and flakey and has lost its usual adherence. Due to the way skin cells fit together, it can peel off in quite large papery, strips, from the site to the surrounding area.
Clever stuff this skin!