Daffyfan2003 said:
How does the neighbor's baby fit in to this?
It was to subtly teach kids that both death and birth are parts of the lifecycle. Earlier in the same episode, they showed a film of a kid visiting his mother in the hospital after she'd just had a baby.
Another observation: Just before that somber scene with Big Bird and the adults, they showed two films with upbeat musical scores: "That's about the size of it", and "Good things growing better, every day". These happy films beforehand appeared to be comparable to the barber lathering the man's face before shaving it. Then after that unforgettable scene, the camera gradually pulled back for a moment of silence, then they went into that water-dripping-from-a-sad-flower film set to Vivaldi's Lute Concerto in D.
Noggin's showing of this episode might not have had all of these sequences included.