Ok, you asked for it.
First off I love this thread!
For us really it's a combination of being immersed in Disney, the convenience and the relaxation factor.
When I go in vacation it usually takes me a good week to really get relaxed to enjoy what I'm doing and not to think of work and other worries. That's pretty bad. However when you go to Disney and stay on site all that goes away quickly. Right away I know that most/all things are taken care of for me. That statement there just relaxes me. I know when I get on the plane on day one I can have just a big sigh of relief. I appreciate it and so do my wife and kids.

That's what a vacation is for.. to relax and have fun, right?
So being on site there's no figuring out transportation from the airport, no renting a car, figuring out where to eat (or wasting money on crummy places to eat), driving to and from the parks and fighting the traffic, feeling that I wish we had of just stayed on site.
It's weird but being on-site gives you this sense of a special community that you're a part of. Again a feeling you cant describe. Knowing that you dont have to go home to a hotel that is far away and is accessed by your car and your driving at the end of a great day is a wonderful feeling.
Our first trip we stayed at the Embassy Suites Lake Buena Vista just off the east gates International Drive. It was a nice place to stay no question. However we felt outside. Walking around our hotel was just not good enough entertainment being that close to Disney. During that trip we were at MK on day and stopped at the CR for lunch. My DW said wouldn't it be awesome to just stay here and not go back to the other hotel?? I looked around and said wow you are so right. This WOULD be awesome. So we switched to the CR for the next 4 nights. WHAT a difference it made. It was SO convenient, so relaxing, so cool! The ability to wake up, walk from your room and 2 minutes later your riding on the monorail to the MK or Epcot. Priceless This trip was kid free as we didn't have any yet and even still it was a tonne of fun.
Our next trip in 2005 was incredible. We were totally immersed in Disney for 12 days with our 3.5 year old DS. We stayed at the Swan this time. The location was fantastic right beside Epcot and the Boardwalk. There were SO many things to do and see without even going to a park. Yes, without going to a park. Talk about immersed.
I remember one night walking leisurely from the Beach Club to Epcot with my DW and DS after eating at Beaches and Cream on a warm summer night. We took our time looking around enjoying the view of the Boardwalk the resorts, the boats, the sand and the water. Everyone we saw had a smile on their face, holding doors for other people and just generally being in a great mood like us. It was something you can't describe but you felt... magic. And, also something you'd never get being off-site.
I know there are people that for some reason dont see the benefit in staying on-site. Its their choice and I respect it. I think though that those people have never actually stayed on-site for more than 2 or 3 nights. From the posts that I see on this board and our experience is that once you stay on site its really REALLY hard to go back to staying at a hotel off-site.
For us (unless it wasn't possible financially) when we stay at Disney it'll always be on-site for sure. I really cant imagine it any other way.