What does the idiom you spread your wings as a new nation mean?

It means that a nation is coming into it's power.

This isn't an idiom. It's a metaphor comparing the nation to some sort of bird just spreading its wings to fly. You can tell the difference because you can figure out the meaning by context - idioms make no sense even if you try to define each Word.
To spread one's wings means to begin to fly, as a baby bird does when it leaves the nest. So "we...spread our wings as a new nation" means we began to govern ourselves, to see what being a new nation was like, how it felt to be a new nation.