Vulcanization, a process that makes rubber stronger and more durable so that it will last a longer period of time, is one of the reasons that rubber cannot merge back together. But scientists have found a way to create a rubber that is capable of healing itself of damage all on its own at room temperature, or even faster in a heated environment.
With rubber being as tough as it is, one thing that it fails to do is fix properly when torn or punctured due to the vulcanization process. This is why it's important to get a tire patch when you get a flat tire, or completely replace the tire if the puncture is in a spot that a patch can't adequately seal.
The new self-healing rubber compound demonstrated in this video is capable of re-bonding to itself and maintaining all of the durability and elasticity qualities that rubber would normally have, even when it hasn't been damaged. The rubber compound has been given carbon and nitrogen additives, which are said to be critical to the reformation process.
To put this into perspective on how it could be a useful idea, imagine having a flat tire and then just being able to park in a garage (near room temperature), and then walking out to a completely mended tire that fixed itself. Magic? - No, just science.