The beauty of natural areas is greatly enhanced by wide biodiversity. Hiking through an old-growth forest clearly pleases the senses much more than through wasteland or parking lots.
A diverse range of species of many sizes, shapes, and colors living together in an environment can be much more inspiring than any work of art. Human imagination is based on things experienced through our senses. When species are lost, the opportunity for future humans to experience first-hand the sights and sounds of these animals is lost. The loss of species on earth decreases quality of life for future humans who will who will have lower variety of species to appreciate.
Some species though may live on through documentation like photos, video, and audio recording as long as the data is maintained. Data decays over time however and it cannot replace the benefit to future peoples of seeing the specie in the context of its living environment.
Humans alive today are temporary inheritors of a fragile system. We have a responsibility to preserve the beauty of natural environments for future generations to appreciate and benefit from.