I would say that we aren’t more “important”, but we are definitely special. So we have a responsibility to look after the planet for all the other animals because we have the power to kill them all, and we have driven species extinct, as well as save them. And we need them to survive ourselves, and we ought to be nice to them anyway.
I suspect you may be asking along the lines of animal experimentation.
There are no easy answers to that question, about why an animal’s life is worth less than a human’s. But really, in real animal testing, the animal isn’t necessarily harmed, and their lives could help thousands or millions of human lives.
I think every animal thinks its more important than the other animals (even if it doesn’t know that its thinking it, because it can’t think…) because of *survival instinct*. Every animal wants to survive and most animals will go to any lengths to do that, by killing, or fighting, or even giving up thier own lives for their children.
If you look at it another way, humans are some of the only animals that value the life of *other* animals over their own! You wouldn’t catch any old monkey staying out all night and risking hypothermia just to make sure a cow had its calf properly would you?
I don’t think that we are, we just happen to have become dominant on the Earth. We are probably the most intelligent animal on Earth, but some experiments have shown that other species, like dolphins and whales are pretty smart too.