For me personally, the question of "why space?" is buried deep in my childhood, when I watched Mercury and Gemini launches on TV and saw the amazing pictures in LIFE magazine. I decided that flying was the coolest thing you could possibly do, and flying in space, even cooler. Long story how I ended up an optical engineer instead of an astronautical engineer, but I never lost the interest in flying and space stuff, and I'm thrilled to see that space is on the national and world agenda more today than in many years.
But I recognize that "normal" people may desire more of an explanation than "it's really cool," and of course it is much more than that. You can list many direct and spin-off benefits of past space programs, and people will accept that weather satellites and velcro are good things. We have those, and we can launch more satellites as we need them, but why do we need more piloted space flights? To go back to the Moon? To Mars? Why spend all that money in space?
First of all, we never actually spend money in space -- there are no boxes full of $100 bills ejected from the Shuttle's payload bay. The money is spent here -- on paychecks, materials, manufacturing, business services, research contracts, graduate student funding, lots of things. It helps people on Earth, generates technology for future products, and stimulates interest in science and technology.
That's my big thing, the kids and the needs of the future -- I'm sure we will always have more than enough rock stars and lawyers, but can we inspire enough kids to be interested in science and technology to meet future needs? Not just for space, for all fields. Anything that can help this is worth it in my opinion. Space should be an international effort, but I would like to keep some of those future jobs in North America, if possible (there's plenty of room for all in the vastness of space, but note that China and India both have very active space programs and apparently associate this with future growth and leadership).
There's much more to say, but I will stop and recommend a web site called The Space Review (see for example this article). Lots of thoughtful articles there, largely by people who favor space exploration, but who are also very critical of what we do and why. There are choices to be made, but taking any of NASA's 1% of the Federal budget and applying it to "more pressing current problems" is not the answer. We should actually invest more money in space now if we hope to continue to get benefits in the future.