Cats arch their back when playing because they are acting out an encounter that may well take place in real life. It is a strong form of body language.
If a cat feels threatened by, for example, a large dog she pulls herself up on fully stretched legs and simultaneously arches her back. She will also take on a sideways stance and the hairs stand on end. The four behaviors make her look as large as possible (an ‘enlarged cat display’) in an attempt to convince the dog that it is confronting an opponent who is best left alone. This display is more successful than running as the latter encourages the chase.
When cats play they are acting out hunting and encounters as described above. So when you see cats arch their back when playing it is probably more likely to be a kitten encountering an adult cat because the kitten sees the adult as a threat albeit in play.
You may also see it when a young cat sees his reflection in a mirror or his shadow on a wall. He thinks he’s encountering another animal who might be hostile towards him and he takes up this defensive/aggressive position making himself as large as possible. It is the result of an instinctive desire to practice a defensive maneuver and it’s enjoyable. Humans do these sorts of things too.
In answer to the question “Why do cats arch their back when playing?” the answer is that they are acting out, as if in practice, a potential real-life encounter with an aggressor.