Another way of explaining what your teacher means is that, while none of these situations is currently happening (you are not now working with animals), that it is something interesting to you. You would enjoy it. You can't say I do enjoy it, because you are not doing it, but you imagine yourself in that situation and that would be enjoyable. Would expresses a possibility.
I know I shouldn't use a word to describe itself, but I hope you'll find it clearer this way?
You are correct. It is not because what is said is 'imaginary or unreal' that 'would' is used in these sentences. We use 'would' when, for instances:
1. expressing a desire or inclination : "I would love to live in London."
"I would enjoy working with animals. I think being a veterinarian would be rewarding."
2. expressing a polite request : "Would you open the window, please?"
3 expressing a conjecture, opinion, or hope, as in :
2. Working as an architect would be really fascinating.
3. They would love directing a tv show.