Paul here seizes on just one verse from Isaiah's memorable prophecy in order to prove that Jesus was given a special mission to take God's saving message to Gentile people as well as to members of the Jewish race. He was having a hard task in convincing some Jewish Christians that he was right about this, and Isaiah's words, 'The root of Jesse shall stand as a signal to the peoples', seemed to lend powerful support to his argument.
Of course, then as now, some Christians probably said that Isaiah had been talking about the people, or the kings, of Israel and Judah, not about Jesus. Paul will have none of it. He asserts that whatever was written in former days was written for our instruction, to give us hope and encouragement on our own faith pilgrimage. This doesn't mean that the prophecies of Isaiah and others didn't have a different meaning at the time, only that they have a special meaning for Christians too, and that meaning is just as valid.
The special meaning of the passage for Paul is that people of different races are meant to live in harmony, just as the wild and domestic animals are supposed to dwell in peace together.