I suggest reading through the 'skills and interests' section here. Then try to 'sell yourself' based upon those qualities:
http://www.learndirect-advice.co.uk/helpwithyo urcareer/jobprofiles/profiles/profile169/
e.g. the answer to the question you're worrying about might start something like this:
"Obviously I love animals but I also enjoy working with people. I know that part of the job is about building relationships with the animals' owners and I'm sure I'd be good at that part of the job".
It might also be a good idea to get across the message that you're the type of person who can stay calm in a crisis and 'step back' from emotion. A veterinary practice needs staff who aren't going to scream or faint when someone rushes through the door with a dog, which has just been hit by a car, covered in blood and whimpering in pain. Equally, they need someone who isn't going to burst into tears every time an animal is put down. (However, they do need someone who can comfort the owners).
Stress that you understand about the 'down side' of the job but make it clear that you're well-equipped to cope.
Chris