Also the final reprise of "Rainbow Connection" where the camera pulls back to show most, if not all, of the Muppets created by Henson at that point singing together and celebrating Jim Henson at his crowning time of success.
What really does it is that it happens right after the movie set got trashed, destroying all their hard work and, had the ending not been a Foregone Conclusion, would have turned it into a Shoot the Shaggy Dog moment. Then, when everybody looks like they're at their absolute peak of despair and about to give up, a real rainbow appears through the gaping hole in the ceiling, the reprise of "Rainbow Connection" plays, and everybody starts singing. Dawww...
...Thanks to the lovers...the dreamers...and you!
And just before the above, everybody working to get the set ready, not to mention the performance of "The Magic Store." Seeing everyone psyched up to get started after their epic journey was seriously encouraging.
Kermit and Piggy's dinner together.
The cameo from Orson Welles when he gives the Muppets their break and their reaction, particularly Kermit's, afterwards. Made even more heartwarming when one considers that Welles's character, Lew Lord, was based on the man who helped Jim Henson into stardom, Lord Lew Grade.
What really makes it is that Welles looks quite intimidating, making you worried that he's going to give them yet another hard time before the Foregone Conclusion. And then he simply tells his secretary to get the "standard rich and famous contract."
Welles also showed that he could act even in a 30 second cameo. Notice the brief hesitation Lew Lord makes before paging his secretary, as the impulse to give them that contract wasn't an immediate one.
Kermit's plea to Doc Hopper when his posse catches up to them in the western ghost town. Even after about half a dozen times Hopper has made it clear he wants Kermit dead, Kermit is nothing but forgiving toward him.
Kermit: Well, I've got a dream too, but it's about singing, dancing, and making people happy. That's the sort of dream that gets better the more people you share with it. And, well, I've found a whole bunch of friends who have the same dream. And...and that kinda makes us like a family.