Many people dismiss space exploration as a luxury, but this attitude is not only wrong, it's dangerous.
Satellite technology has revolutionized our planet in almost every way. Weather satellites help us track developing hurricanes, allowing meteorologists to warn people days in advance. That saves thousands of lives. Communication satellites allow us instant access to information all over the world using radio, television, and phones. Some people credit in part the fall of the Soviet Union to ease of information access; the people in those countries saw what the rest of the world was doing, accelerating the process of reform. GPS satellites allow us to track ships, airplanes, and even people who may be lost or in need of help.
Reaching outward into space has helped us in other ways as well. A fleet of satellites (YOHKOH, SOHO and others) study the Sun so that we can better understand it. Huge solar eruptions can damage satellites and cause power blackouts (like in Quebec in 1989), and the Sun directly influences our environment. Understanding the Sun is a critical use of space technology.
Exploring the other planets helps us put the Earth in context. Why is Mars dry, cold, nearly airless, and dead? Why is Venus covered in thick clouds and suffering a runaway greenhouse effect? Why do hurricanes on Jupiter last for centuries? All these questions (and thousands more) help us understand our own planet, and allow us to see how humans are affecting it. Certainly understanding asteroids is important-- we need to learn how to move them in case one is heading our way; an asteroid impact could wipe out all humans on Earth, so our very future is tied to space travel.
There are simple technological reasons for space exploration as well. Some estimates say that for every dollar invested in the Apollo program, more than 20 have been returned. That's a huge payoff! Computer tech, communications, rocketry, and many other fields have benefited hugely from space exploration.
And there is one more reason. Humans strive to learn, to explore, to push boundaries, to see what's around the corner. This is in many ways a fundamental need, and space exploration is a fantastic manifestation of that. The Universe is huge, beautiful, mysterious, and, ultimately, knowable. Even if the other reasons were not there, this alone should be enough for us pursue our exploration.