As the story goes, several events take place that are directly shaped by Hezekiah's prayers (read 2 Kings 18-20 for the full stories). Each time Hezekiah prays, the Lord directly answers, and some times in extremely miraculous ways. The coolest example is when King Hezekiah gets sick. The prophet Isaiah (yup, the same Isaiah that has that book after the book of Psalm) comes and tells Hezekiah to get his house in order, he's gonna die. Hezekiah is so distraught at this, he cries out to the Lord in prayer and asks God to remember all the good he had done in his life. God grants Hezekiah his request and gives him 15 more years to live. As a sign that God is going to do this, he makes the SUN MOVE BACKWARDS. No kidding. How cool is that? To see the sun move backwards (Hezekiah sees the shadow on the steps of the palace moving in the opposite direction.)
So Isaiah comes back and asks Hezekiah what he had done. Hezekiah, probably with a little pride in his voice, tells Isaiah he showed Babylon everything he had, ahhhhhh, to be so rich. Ha! He showed them who was king of Judah! Isaiah more or less tells him that while Hezekiah thought that this display of wealth would impress the Babylonians, all it did was show them what the kings of Judah had, and what they could get from them. One day the kings of Babylon would come and take it all away.