If we put aside the desires that humans feel for obtaining and possessing animals, there are still many deeply ingrained reasons for why animals are a necessity to our planet, not a luxury.
Firstly, Earth consists of billions of fragile and interdependent webs of life which wholly rely on each other for continuation. The interaction of an organism with its environment are vital to its survival and that of the ecosystem. For example, to a casual observer, the Bee (or Anthophila) may appear to be a cordial little black and yellow insect which delights in vibrant flowers and warm summer days. While this may actually be true, bees signify much more than that. The pollination of flowers carried out by bees is directly responsible for 70 percent of the fruits, vegetables, nuts and seeds that we consume on a daily basis. So, if bees became extinct, which seems like an extremely probable outcome at the moment, then half of the products on our supermarket shelves would simply cease to exist unless we hastily recreate an equally effective pollination method.
Furthermore, mimicking complex natural architecture has provided us with an evolutionary shortcut. We have based much
of our science progress on structures such as a spider's web, a bird's wing and the architecture of ants. This is called biomimicry. So through this inspiration which we draw from animals, we have developed medicines that save millions of human and animal lives annually and moreover, we have advanced novel technologies.
There are many reasons for why we not only want- but also need - animals.
As said by Donny Fernstrom on Quora 'the loss of ants alone would result in near-global devastation of all land-based ecological systems. Only antarctica and the arctic would be left out.'
Conclusively, even if we completely overlook all of these pragmatic technicalities, surely you can see the need for animals?
The majority of children's books are based on or around the idea of animals. Surely this alone highlights our great devotion to the species that we cohabit the planet with... They provide us with great enrichment and beauty. The power of an eagle's flight... The fragility of a butterfly... The tenderness of an elephant's familial bond... Would our planet not be so insipid and characterless without these things?