Actually, funding for National Parks and National Forests is pretty minimal, considering the mission - no money "pours in", ever (I've worked for one). We have them because of legislation like the Weeks Act of 1911, whose pragmatic purpose was to preserve watersheds and prevent flooding. Tourism came later.
These areas are the only practical way to preserve entire functioning ecosystems intact; we are only just learning the benefits of that. Did you know that several significant bacterial species have been discovered in the hot springs at Yellowstone, for instance?
Allowing tourists to visit offers people the chance to experience nature in its true glory. In a society that is increasingly detached from the natural world, this is a major thing. People need to feel a sense of ownership in order to feel a sense of responsibility. They need to feel a sense of responsibility in order to play their part in creating a sustainable society. If we don't create a sustainable society, our species is probably going to go extinct.