There are a great many pokemon who possess the capacity to heal themselves. There are very few who can also heal others, and of these only Chansey (and its rare evolved form of Blissey) can do so without sacrificing its own health in the process. Sadly, for such a good-natured pokemon, this has made Chansey for much of their history little more than a target. They were once a common pokemon, but became rare from warfare; generals conscripted them into their armies and when they could not guarantee their capture they killed all they could find to deny them to the enemy.
It was these massacres which provided the impetus for the first pokemon centers, started by idealistic trainers who gathered the local Chansey and used them to heal the pokemon of anyone who promised not to use their pokemon to harm others. In time, these became an attraction so enormous they were funded by the state – not for military use, but so people would have an incentive to visit their capitals and be able to hold battles when they got there. And as the centers grew, the already popular sport of pokemon battling exploded in popularity, while an age of (relative) peace spread across the land.
In modern times, others have wished for the powers of Chansey, and often these wishes have come true; some have even turned the art of the wish into a healing science almost as powerful as a Chansey's egg. Modern medicine based on the usage of healing machines has in some ways made Chansey obsolete, but they are expensive to maintain. Indeed, many Chansey still work in the centers, operating the machines or just healing wounds themselves.