For me, it was the feeling of hope in the future the show imparted. In real life in the 60s and 70s we were living in the nuclear age and the doomsday clock was clicking steadily towards oblivion. This show demonstrated humanity getting past that point and surviving the insanity.
A lot of people talk about the influence the show had on scientific areas (technology, medicine) with all the futuristic gadgetry (communicators, replicators, transporters, etc.) but when I stop to think about it, I wonder if it's the concept of IDIC that really made an impact on us, nerds or not.
Sometimes I wonder if Star Trek may have saved the world.