What a great question. More and more schools are asking this question and pointing to arts programs for "simple" budget cuts in order to spread the budget around. The arts are incredibly important for all students and humans.
First, arts include all sort of creative expression. Without a creative outlet, humans become resentful and irritable. Painting, sculpting, building, singing, dancing, writing, and creating are all valid expressions of thought. Just as a journal/diary can help a person organize thoughts through a tough time in his or her life, so can building a treehouse, writing or singing a song or painting a portrait.
Second, arts of all kinds provide beauty in our world. Without beauty, there is no pleasure. Without pleasure, there is only discontent and anger. People need beautiful things to look at, experience, and enjoy. True, not everyone see the beauty in a piece of modern art or a mural in an inner-city park, but the beauty is there for the creator and his/her audience.
Third, without arts programs in elementary, secondary, and post-secondary schools, how will students discover these talents? Many students don't find their niche or the thing they are best at in life unless given the opportunity to explore with paint or other medium in art class, or wood and metal in shop, or material and thread or food in Home Ec. We need artists in our world, and they need to know their talents.
Fourth, the arts encourage problem solving. How many times have you had a project in mind and ran into an issue that had to be solved before the final product could come from it? Change the medium, or the type of paint or glue, use a different kind of material, or some other issue. It's great for brain development!