Star Trek inspired me by showing me a galaxy full of equals, and a galaxy where you don't interfere with indigenous cultures. A federation of worlds where everyone's needs are taken care of and human rights are respected.
And most importantly, an imperative to talk things out in a hostile situation before shooting.
For me personally, two shows/movies made a big impact on me as an early teen:
Star Trek (TNG at the time)
Jurassic Park
The reason being that most of the characters in those two shows/movies weren't superheroes/cops/etc. They were geeks and nerds.
Star Trek, the geeks and nerds are prized in society, and their purpose is to explore, to learn. They were scientists. They didn't accomplish their thing by conquest, but by diplomacy and science. There, of course, were battles/warriors/etc, but those are shown as the exception.
In Jurassic Park, every character is a nerd (the Dino obsessed boy, the Unix hacking girl, the Paleontologist/Paelobotanist, the Geneticist, Mathematician, etc). The good guys AND the bad guys were nerds. No strong-man was needed to save the day.
I was a nerdy kid, and those spoke to me. I now work in a science research lab, and love it. I'm still a nerd.
Putting my mod hat on here- This post is about how Star Trek inspired you, not what you think of Whoopi Godlberg. Think of the quote as why Star Trek inspired someone, not who that someone is.
We-ell, it inspired me to respect Star Wars more. Hard to take seriously people laughing about midichlorians and space wizards, and at the same time being serious about transporters and creating matter from nothing and all that "post-scarcity" stuff.
Also Star Trek: Bridge Commander is an awesome game, simpler than something like Battlecruiser Millennium, but still cool. It's the Star Wars game I'd like to see minus Star Wars.
I know this is meant as a more serious thread, but all I can think of is this quote from futurama...
It taught me so much. Like, how you should accept people, whether they be black, white, Klingon or even female... But most importantly, when I had no friends, it made me feel like maybe I did.
For me, I would say I just have a lifelong interest in space and a desire to know what's out there.