OK, so most people seem to think the quote is about Scorsese's film professor. However, what does the quote mean in the context of the story? How does it fit in with the rest of the film?
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The quote at the ned of the film is as cryptic as it is an ineffesctive device to close the film.
I think the significance of the quote is solely meant for the filmmakers.
(a) La Motta learned to accept those qualities in himself which he previously felt uncomfortable with. He found a healthy way of showing that "I'm the Boss, i'm the Boss" through his stand up work.
(b) It's ironic; he didn't find the path to wisdom because he needed to recite quotes from films like "On the Waterfront". A example of his low self-esteem or inauthenticity and need to prove he wasn't an animal.
Either way, it's a hopeful resolution as it suggests more self-acceptance.
"The royal penis is clean."
So, for the second time, [the Pharisees]
summoned the man who had been blind and said:
"Speak the truth before God.
We know this fellow is a sinner."
"Whether or not he is a sinner, I do not know,"
the man replied.
"All I know is this:
once I was blind and now I can see."
Nothing cryptic here. What the once blind man is telling the Pharisees (who accuse of false pride and blasphemy) is that he is not here to judge the man Jesus; all he do can is put forth what the man has done for all to see (he claims to have been healed by the man), which may stand or fall on its own rights.
This fits nicely with the theme of the film - that of a flawed human who simply does as he does. There is no tone of either judgment or idolization of LaMotta. What we get is a beautifully cinematic ride into the psyche of the protagonist. Everything else is up to the viewer.
It seems pretty obvious to me. I've just had a search for various other theories on the meaning but they all are needlessly complex and silly, imo.