I love The Original Series, but I find I have to watch it with an active imagination. It leaves a lot of blanks for my mind to fill in - like a good book.
If you're the sort of person for whom everything needs to be displayed explicitly onscreen and every second has to be packed with hyper-stimulating audio and visuals, then TOS is probably not the right series for you.
^^That's a pretty good summary of how I feel about TOS.
With TOS getting a very nice HD transfer back circa 2006, the live action has never looked better. The bold colors used (especially in the first season) look great.
Another thing about the '60s is that we have '60s influences in pop culture now, Mad Men is super popular, so I actually think TOS has aged reasonably well. Conversely, the first two seasons of TNG reek of tacky '80s fashion which have not aged as well, possibly because not enough time has passed. I suppose it's possible in another 25 years, the 1987 look will be respected retro style, but it doesn't seem like it.
TOS is easy to get into and enjoy, it's usually pretty clear what's going on in each episode and each is pretty self-contained. The trio of Kirk, Spock and McCoy is also very easy to understand and enjoy, their clear respect and affection for each other comes through in most episodes.
As many others have already said, TOS also is a product of it's time, and as such if you understand the themes and general pop culture of the time, you can see exactly how TOS fits in. The cold war allegories, the western influence (Gene Roddenberry wrote many episodes of
Have Gun Will Travel just before creating Star Trek), the military influence (many of the production staff had served in WWII), etc.
I grew up loving classic TV, I loved watching Get Smart, Dragnet, Lost in Space, I Love Lucy, Bewitched, etc. If Nick-at-Nite showed it in the early-'90s I probably watched it. So for me it all fits in with TOS and it just clicks. I also love TNG, but for different reasons and in different ways.