I would say that this is true because the more you know about something and teh more you think about it, the more you see that it is fundamentally imperfect. You cannot really be very familiar with something and still think that it is perfect.
This can be seen in relationships -- you can't live with someone as an adult for very long without knowing their faults.
This can be seen in teaching. You cannot, in my opinion, be a teacher for a long time without understanding that there are things that teachers claim to be able to do that they are really not able to do.
So I think this quote is sort of related to "familiarity breeds contempt."
You may have heard people say, "Don't ever make what you love your life's work." The idea here is that you will grow to hate it. Every industry has corrupt people or flaws within their systems. With age or time invested in an industry, these problems become more perceptible. Thus, staying out of it will help you keep loving it.
Another way you could look at the quote you are analyzing is this: the longer you work at a job, the better you know it. Then, when you see failure or poor performance around you, it gets easier to judge it.
This might not be the right answer because I am guessing but I hope you will consider it.
I believe it means that each person has their own independant skill that they can master but they are not aware of it or they do not belive that they have that skill.
I think all these answers are good and touch on possibilities. Shaw's assertion overreaches, however, when he says "every." Having mastered several usual and unusual professions myself, now that I'm old I don't think it applies to every profession. His assertion is also rather peculiar in that "skeptic" is a slippery and often misunderstood word: it does not mean "cynic," one who discounts new ideas or thinks the future must turn out badly, as is often assumed.
As Shaw was British, I consulted the OED to give a better idea of the sense of the word in his mind. Concise Version OED 2008 states: a person inclined to question or doubt accepted opinions. Ah hah!
In one's own mastered profession(s) I find that one can question or doubt:
-it's real value to others and was this value been overrated?
-it being as satisfying at the end of the years, or as glamorous as one imagined, when younger and starting out. It is probably disappointing in several respects, as many things turn out to be with time. As has been said, in the beginners mind there are many possibilities; in the expert's, few. And most of us habituate to many repeated exciting stimuli and experiences, they become humdrum. Or years of sex with the same person. A tour guide at Niagara Falls or other amazing attraction will not experience the excitement or amazingness after years of seeing it again and again. This is how most of us are wired.
-Now it is up to you to also think of counterexamples. We've all heard the old saw "Nobody on his deathbed says 'I wish I had spent more time at the office.'" Well, how about a medical researcher such as Dr. Jonas Salk, who came up with the cure for polio? Do you think he was questioning or doubting his or his work's value?
I have mastered two specialty professions and skills which I now volunteer at my local animal shelter. Am I skeptical in any sense whatsoever about them? No.
As with many quote originators, including myself, my guess is that Shaw came up with this one by discovering it in himself.