Fraser at Universe Today had a good thought -- he (like any space enthusiast) gets asked about the value of space exploration all the time. So why not post an answer, and ask others to send in answers as well?
So he sent an email to a few folks asking them to answer this question: "Why should we be spending money exploring space when there are so many problems here on Earth that we need to solve first?"
He's been posting the responses, and they've been really good. Here's mine... but you should go over to UT and check out what the others have said. Lots of ammo there for the next time a luddite wants to cut NASA's budget!
First, the question of why spend money there when we have problems here is a false dichotomy. We have enough money to work on problems here and in space! We just don't seem to choose to, which is maddening. $12 million an hour is spent in Iraq; the US government chose to do that instead of fix many problems that could have been solved with that money. NASA is less than 1% of the US budget, so it's best to pick your fights wisely here.
Second, space exploration is necessary. We learn so much from it! Early attempts discovered the van Allen radiation belts (with America's first satellite!). Later satellites found the ozone hole, letting us know we were damaging our ecosystem. Weather prediction via satellites is another obvious example, as well as global communication, TV, GPS, and much more.
If you want to narrow it down to exploring other planets and the Universe around us, again we can give the practical answer that the more we learn about our space environment, the more we learn about the Earth itself. Examining the Sun led us to understand that its magnetic field connects with ours, sometimes with disastrous results... yet we can fortify ourselves against the danger, should we so choose. Space exploration may yet save us from an asteroid impact, too. Spreading our seed to other worlds may eventually save the human race.
But I'm with Fraser. These are all good reasons, and there are many, many more. But it is the very nature of humans to explore! We could do nothing in our daily lives but look no farther than the ends of our noses. We could labor away in a gray, listless, dull world.
Or we can look up, look out to the skies, see what wonders are there, marvel at exploding stars, majestic galaxies, ringed worlds, and perhaps planets like our own. That gives us beauty and joy in our world, and adds a depth and dimension that we might otherwise miss.
Space exploration is cheap. Not exploring is always very, very expensive.