There shall come forth a Rod from the stump of Jesse, and a Branch shall grow out of his roots…And in that day there shall be a Root of Jesse… (Isaiah 11:1,10)
In 2 Samuel 7, after King David is confronted by the Lord for his adultery and murder, God does something ridiculously gracious: He promises David that He would not only create a dynasty of kings from David, but would also raise up a Ruler from his lineage, from the tribe of Judah, who would establish an eternal Kingdom that would endure forever! From that point on, when the people of Israel waited for the Messiah, they expected Him to be a King who would be descended from David.
But as the centuries passed and Israel’s kings got more corrupt and evil, the prophets warned the people that God’s judgement would come on the house of David (his royal dynasty), using the imagery of an axe ready to cut down the useless tree. Israel didn’t listen, and God’s judgement did indeed come in the form of the Exile, where the king of Israel was dethroned, and it seemed like the promise God made to David had been taken back; there can’t be a King from the line of David if there’s no more kings at all, right?
The prophet Isaiah saw this beforehand (cus, you know, he’s a prophet), and says in Isaiah 11 that the line of David would be like a tree stump that had been cut down in judgement. But, a Rod (a Ruler or King) would come from this seemingly dead lineage; a new Branch would grow out of the stump! Although the promise to David seemed forgotten, and the royal line of kings was cut short, God would still preserve his descendants throughout the generations in order to raise up the Messiah at the right time! He refers to it as Jesse’s line instead of David’s because Jesse was David’s father, implying that there would be a fresh start, a back-to-origins kinda thing.
So, the phrase Root of Jesse or the Branch from the stump of Jesse is a title for the Messiah, Jesus! It refers to how Jesus is a descendant of David, and is the fulfillment of the royal dynasty that was cut down in judgment. Even though there were no more kings from the line of David after the Exile (the stump), Jesus the Messiah still came, and was from his line (the new Branch). Against all odds, when it appeared as though there was no way for God to bring His plan to pass, He knew what He was doing, and the Son of God came, changing the world forever. I hope this made sense, feel free to leave any questions or comments you might have!