So was it when my life began; So is it now I am a man; (3-4)
These lines show us that the feeling the speaker has for rainbows continues through time. We can imagine that lots of other things in his life have changed since his childhood—perhaps he's finished school, gotten a job, gotten married, had kids. He may have worries that he could never have dreamed about when he was a child. (It happens, Shmoopers.) Yet, with all the changes time brings, he continues to rejoice in the beauty of rainbows.
So be it when I shall grow old, Or let me die! (5-6)
The speaker forecasts that his love of rainbows will continue even as time continues to pass. His kids may move out, he may retire, heck, even his wife may die, but if he ever stopped loving something as beautiful as a rainbow in the sky, that is when he'd want to die, too.
The Child is father of the Man; (7)
This line reverses the normal direction of time. Men are fathers of children, right? Right. But think about the word father as just meaning someone who came before. Every man was once a child, so in a way, all men father (or form) themselves. Time changes many things, but that child is still inside somewhere, perhaps briefly coming out to play when a rainbow appears in the sky.