Having right of abode means you’re allowed to live or work in the UK without any immigration restrictions (ie you won’t need a visa to come to the UK and there’s no limit on the length of time you can spend in the country).
All British citizens automatically have right of abode in the UK.
Some Commonwealth citizens may also have right of abode.
You can prove you have right of abode if you have a UK passport describing you as a British citizen or British subject with right of abode.
Otherwise you’ll need to apply for a ‘certificate of entitlement’.