That sounds like a job with little chance of making money.
But you have to become a lawyer first.
That will take a lot of money and quite a few years in school but there are lots of universities where you can study law..
Then study case law about animal rights. Then offer your services by advertising that fact. One could volunteer their services to animal rights groups. Most will have no money to pay you.
It might be better to study to become a para legal. I think most lawyers that do animal rights cases do it as a kind of charity because there is no money to be made doing it and it costs a small fortune to become a lawyer.