Inspiration is a tautological goal. "Why do we do X? Because it is inspiring!" only works if we find X worthy for some reason. How about: real goals that substantively improve life on Earth?
I'm okay with that. You don't have to be inspired by it. I am, and I think many other people are as well.
Human civilization has a lot of problems. I don't know how to solve them. I believe that demonstrations of human ingenuity, capability, and technological aptitude have the potential to cause people to aspire to achieve something, maybe to solve "real problems".
The best part is that they aren't mutually exclusive. I'm all in favor of improving life on Earth too, and I'm interested in hearing some of your ideas of ways to solve the problems we have here.
People who try to study the evolution of well-being (for various definitions of it) show that it stagnated (best case) or started dropping with the generations born in the 60's and 70's. So obviously nothing improved life in those years and decades which follow space exploration.
In addition, because I believe it is worthwhile to attempt difficult things to see what we can achieve. Inspiration is a valid goal.