Genesis 1:8-9

Then God spoke to Noah and to his sons with him saying. Now behold, I am establishing My covenant (binding agreement, solemn promise) with you and with your descendants after you.

God made a covenant with Noah, their descendants, and the animals that He would never destroy the world again with a flood.  When God makes a covenant with someone, it means that He will never break His promise.  God watches over His Word and His promises and makes them come true.  A rainbow is a sign to this day of that promise to this day.