First, after having launched humans in space, the space station was considered the next logical step in space.
Why? Because the space station is the best way to allow humans to stay in space for any length of time or even to occupy space permanently.

Second, the space station is halfway to anywhere as the most difficult part of any spaceflight is the trip from the ground to low earth orbit.
Even astronauts for the Moon or Mars will be spending a few minutes in earth orbit.

Third, the space station is low enough to allow rockets normally meant for commercial launches to carry astronauts (if they chose to) to it.
This means other nations can participate and bring their own reasons for spending time in space. This alone can make the space station a much more sustainable project than a Moon or Mars trip.

Fourth, the space station is there and can welcome crews. This means if we wanted to, we could change its purposes, its owners. Mir has known transformations too, while the Soviet Union collapsed

Fifth, it's a test: can we leave low earth orbit and drop the space station and undertake manned flights elswhere? I very much doubt so.
If there's no interest in having astronauts in a space station, there won't be any interest in having them on the Moon or Mars either.

Sixth, the space station is a destination and a close one!! That was one of the shuttle's weaknesses, before, it didn't have any place to go but space. Now , it is a destination and not only for the shuttles, but also for Progresses, Soyuzes, ATV's, HTV's, Orions...
Being a destination means possibilities to grow, change, adapt, share, etc...

And seventh, it's a brilliant point of light crossing the sky and inspiring all those looking up.