The rainbow is a sign of God's promise never again to destroy the earth with a flood.
After the flood, GOD promised to Noah that He would never again destroy the whole world with a flood. GOD gave the rainbow to humanity as a sign of this promise for all future generations.
11 Thus I will establish My covenant with you, and never again shall all flesh be cut off by the waters of the flood; nor shall there ever again be a flood to destroy the earth.” 12 Then the Lord God said to Noah, “This is the sign of the covenant I am making between Me and you, and every living creature with you, for perpetual generations: 13 I am setting My rainbow in the cloud, and it shall be a sign of the covenant between Me and the earth. 14 It shall be, when I bring a cloud over the earth, the rainbow shall be seen in the cloud; 15 and I will remember My covenant between Me and you and every living creature of all flesh; the waters shall never again become a flood to destroy all flesh. 16 My rainbow shall be in the cloud, and I will certainly remember the everlasting covenant between God and every living creature of all flesh on the earth.” 17 Again God said to Noah, “This is the sign of the covenant I established between Me and all flesh on the earth.”
(Genesis 9:11-17 OSB)