According to the program, Bruno was found guilty
of questioning the Holy Trinity and the divinity of Jesus Christ, of believing that God's wrath is not eternal, that everyone will be saved, of asserting the existence of other worlds.
To answer your first question, What does it mean that God's wrath is infinite?, it appears, based on the quote from the program, that Bruno was a universalist. This was, and often still is by many Christians, considered a heresy.
The answer to your second question, What is its relation to existence of life on other planets? is that there is none. These were separate, distinct charges, unrelated to each other. There is no implication that the believing in other worlds is in any way related to the believing that God's wrath is not eternal.