God establishes what He has done to reveal what He will do: First Time dove went out and flew over the water its symbolic to Genesis 1 when the spirit hovered over the earth dark and void.
God shows His commitment. Second time the dove brought back a twig (the price of covenant is the very thing God had to do and that is die) Its prophetic God will secure His convenant with the Cross
Third time the Dove is sent never comes back. It is symbolic of when God deploys the Holy Spirit to earth and He never returned back to God. This is a sign of grace.