How do I prove I have the right of abode?

If you are a British citizen, your British passport is itself proof of your right of abode, so you usually need nothing further.

If you are a Commonwealth citizen who qualifies, you cannot simply assert the right at the border; you apply for a certificate of entitlement, which is endorsed in your current passport. Until you hold that certificate or a qualifying British passport, carriers and border officers have no way to confirm your status, so obtaining the certificate is essential before you travel.