For one thing, space is cool. I loved studying space and astronomy as a kid, and really, it's one of the few places where I could see clear (and AWESOME) applications for higher math. I'm an Ohioan, so flight and space are big--we have the Wright Brothers, Neil Armstrong, and NASA Glenn research center in Cleveland.
Over the summer, I visited the Johnson Space Center in Houston. It was a world-class museum, offering something for everyone at rock-bottom ticket prices. I got to touch a moon rock, visit Historic Mission Control, see the Saturn V rocket, and learn that we've figured out how humans could survive on the Moon. It was so neat. If I was a kid, this might have motivated me to work harder in math and science!
Growing plants on the Moon
Space is part of the national consciousness. Star Wars, Star Trek, Apollo 13, and the Challenger explosion are easy examples of space-themed things in our culture. My Tempurpedic bed is (supposedly) based on NASA science.
Sure, it's a government agency, so it's bloated and inefficient. But it does need consistent funding if we're going to reach the Moon again, or Mars. I've been reading Man and the Moon, by Andrew Chakin, about the space program. These moon trips were complicated research missions-- a lot of science was involved, as was the taking of samples and other experiments. It wasn't just, "hey, let's beat the Russians to the Moon."
I don't think exploration can come with set standards. Christopher Columbus wanted to find China; he found Hispanola instead. Alexander the Great wept when he saw there was no more to conquer. Magellean found the way through the treacherous strait that bears his name. Every terrestrial has added to our knowledge about this planet, her people, and her life. (Let's save the arguments about why it wasn't so great for another time.) Why can't extraterrestrial exploration do the same thing? Astronauts are today's Columbuses and Magelleans. They give us license to dream.