Let's not get it twisted.......God is the Creator and Author of the bow. He is it's designer and maker. And when God creates a thing he gives it it's purpose. So, why did he create the bow.
Genesis tells us, that after the flood, God established a new covenant with Noah and his sons. The covenant (or the promise) God made with them was that he would never destroy the earth by the waters of a flood.
Genesis 9:11 And I will establish my covenant with you; neither shall all flesh be cut off any more by the waters of a flood; neither shall there any more be a flood to destroy the earth.
And for a token of this covenant, God made a bow, the bow was to remind Noah and his sons and their sons after them of this covenant God made with man and the earth.
Noah's sons holds the seeds of life within them that will reproduce generation after generation, Each son will reproduce another generation in the earth that will see the bow in the clouds and remember the covenant God made with man and the earth. It's because of Noah's seeds in the earth that we are able to see the promise of God.