I have several reasons for loving Star Trek. Most of them really only apply to The Original Series(TOS) as it is my favorite and the standard by which I measure all of the others.
My first reason is the most "real".
I love Star Trek because it presents a positive outlook for the future. Unlike most fiction about the future this one doesn't show us recovering from a nuclear war or some other disaster which has left the world in ruins. It says this is what we could be. This is our potential. We could be this wise. I think it's really nice to see a vision of the future that's hopeful and gives us something to aim for.
Now my other reasons are far less heavy.
The characters in TOS are great, I mean who can't love the whole Kirk-Spock-MCcoy trio? They are pretty much the greatest trio in television history.
The others series have good characters as well, especially TNG and DS9 in my opinion.
My favorite episodes of TOS are two of the most loved ones, The Trouble with Tribbles and A Piece of the Action.
One of the most memorable ones is And Let That be you Last Battlefield. It had a really poignant point. You have to remember when the show was made. It was highly progressive in many respects.
That's why I love Star Trek.