This, to me, is a silly question. It is so obvious why we need to save other species. The biodiversity of our planet is so beautiful, wonderful, interesting... simply knowing that we share space and time with such amazing species is enough for me to want to keep them there.
We are but one species. If we lose other species we will be alone, and we evolved amidst the greatest biological diversity the world has ever seen. Without them, much like in solitary confinement, we would not cope.
We need other species. Think how much better you feel when you have been out in nature. Science proves that it helps our health. Ecosystem services such as crop pollination is provided to us for free. Medicines too, many of which I'm sure haven't even been discovered.
But most importantly, we should conserve other animals because they are living, breathing beings, with a right to exist. They deserve to live their lives in peace. Who are we to deny them that?