The significance of this is that God has chosen a new King to run Israel, the way He wants it run. And he chose a man that no one thought would be the one, the one that Jesse says "was insignificant".
God chose a man after His own heart.
The significance is to show that when one joins with God, He equips us to do what we need to do, however, whenever, wherever. In the case of Israel now, there was trouble with the Philistines, big and powerful people and how do you fight people like that? with power! God's power!
And also power to rule as the new King seeing there was a King set in place already and it was dangerous for david and Samuel.
His success comes from his obedience to God, power bestowed upon him. Just like the previous King he had God's power upon him until he disobeyed and started doing his own thing.
In order to be successful, we need to be obedient and stay within the spirit of God, His power bestowed upon us as His disciples.