For a story to be so universal, it must echo experiences that happen to all of us. What happens to all of us? The obvious candidate is: growing up. Glen C. Strathy explains it quite well here:
It's also the formula for a rite of initiation, which may be why the monomyth is so popular among young adults. It echoes the process all teenagers go through as as they transform from children into adults. Seen from this perspective, the three stages might be described as ...
- As the young teen leaves childhood, he/she moves towards greater independence from parents.
- Adolescence and the struggle to become an adult (to come to terms with adult power, shoulder adult responsibilities, become sexually active, achieve a new understanding of one's parents, find one's place in the world, etc.).
- Settling into adulthood (marriage, parenthood), so the cycle can continue.
Transitioning from dependence (Mom! Could you do my laundry for me today?!?!?) to independence is hard. And we can all relate to it.
But you asked for What Campbell himself said in The Hero with a Thousand Faces (emphasis mine):
It has always been the prime function of mythology and rite to supply the symbols that carry the human spirit forward, in counteraction to those constant human fantasies that tent to tie it back. In fact, it may well be that the very high incidence of neuroticism among ourselves follows from the decline among us of such effective spiritual aid. **We remain fixated to the unexercised images of our infancy, and hence disinclined to the necessary passages of our adulthood. ** ... The psychoanalyst has to come along, at last, to assert again the tried wisdom of the older, forward looking teachings of the masked medicine dances and the witch-doctor-circumcisers; whereupon we find, as in the dream of the serpent bite, that ageless initiation symbolism is produced spontaneously by the patient himself at the moment of the release. Apparently, there is something in these initiatory images so necessary to the psyche that if they are not supplied from without, through myth and ritual, they will have to be announced again, through dream, from within-lest our energies remain locked in a banal, long-outmoded toy-room, at the bottom of the sea.
As far as a citation for Campbell saying that classic myths from many cultures follow this pattern, he said, again in The Hero with a Thousand Faces:
The wonder is that characteristic efficacy to touch and inspire deep creative centers dwells in the smallest nursery fair tale-as the flavor of the ocean is contained in a droplet or the whole mystery of life within the egg of a flea. For the symbols of mythology are not manufactured; they cannot be ordered, inveted, or permanently suppressed. They are spontaneous productions of the psyche, and each bears within it, undamaged, the germ power of its source.
What is the secret of the timeless vision? From what profundity of the mind does it derive? Why is mythology everywhere the same, beneath its varieties of costume? And what does it teach?