It's a great episode (2096),and it thankfully was included in the "Unpaved" series. Mr. Hooper had passed away by this time, but virtually every other grownup on the street at the time was there.
It's hard to describe the moment. Essentially, the human cast is shown staring in awe and wonder at something. Then we cut to a long shot to show that the people are staring at Snuffy. They slowly walk up and begin patting Snuffy. Almost as if to say "Welcome" and "Are you really real?" at the same time.
Big Bird comes in triumphant. "At last! I told you there was a Snuffle-Upagus, but you didn't believe me!" After the grownups get over the shock, Susan speaks.
Susan: "Big Bird, you have a right to be angry."
BB: "I do?"
Susan: "Sure. After all this time and we didn't believe you - that must have been really hard for you."
BB: "Well, yeah, it was."
Susan: "Well Big Bird, from the bottom of my heart, I want to apologize."
Bob: "Big Bird, from now on, we're going to believe anything you tell us. We promise."
Snuffy: "Maybe we should get that in writing."
The show ends with Big Bird formally introducing everybody to Snuffy and a "group hug" with TV host Phil Donahue joining in. (He was on the street earlier getting his toaster fixed).
The backstory is that this scene was partly written to adress the issue of abuse and how important it is for children to know that they will be believed if they need to be.