I suppose it would be unromantic to point out that ornithologists think that the main reason why birds sing is to protect their territory. The proverb isn't concerned with, or its authors were ignorant of, scientific accuracy.
Even if the sentence had been written this morning after breakfast, I don't believe that scientific knowledge has anything to do with it. It's poetry, gentlemen, and a bit of pop-philosophy. Nothing more, nothing less.
Interesting that someone has posted this question here, as I've been using this quotation as my wordreference signature for some time now. I learned it as a quotation by Maya Angelou, not as a proverb.
To me, it means perhaps more than I can words here at this time.
But for a minimalistic answer, I would say that, to me, it means that I do what I do because I am living my truth — not because my truth is more accurate than anyone else's truth.
And in this sense, birds are, in fact, living their truth when they sing. They are being who and what they are.