The source of that aphorism is probably William H Whyte and is better understood with its context about listening.
"LET US RECAPITULATE A BIT: The great enemy of communication, we find, is the illusion of it. We have talked enough; but we have not listened. And by not listening we have failed to concede the immense complexity of our society–and thus the great gaps between ourselves and those with whom we seek understanding."
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George Bernard Shaw probably didn't say that, "The single biggest problem with communication is the illusion that it has taken place." https://quoteinvestigator.com/2014/08/31/illusion/
The source of that aphorism is probably William H Whyte and is better understood with its context about listening.
"LET US RECAPITULATE A BIT: The great enemy of communication, we find, is the illusion of it. We have talked enough; but we have not listened. And by not listening we have failed to concede the immense complexity of our society–and thus the great gaps between ourselves and those with whom we seek understanding."