This is a great question. There are two main points to Lucretius' poem. First, Lucretius seeks to free people from the fear of death. Second, he seeks to free people from the fear of the gods.
With this stated, Lucretius offers three arguments to persuade his audience. First, he argues from the point of view of atomic theory. In other words, he gives a physical explanation of how the world works to disabuse people from any kind of fear. The assumption is that people are less likely to fear something that they can understand. Lucretius famously says in 1.158:
Nam is de nilo fierent, ex omnibus rebus omne genus nasci posset, nil semine egeret. "For if things emerged from nothing, then all kinds of things could come from all things, nothing would want a seed."
Second, he offers empirical evidence to argue against what people consider the acts of the gods. In other words, the gods do not really interfere in the lives of people. For this reason, they do not punish people. Hence, people should not fear the gods.
Finally, he offers examples of religious convictions, which have gone awry, to show the dangers of following in the logical footsteps of traditional religion. In short, Lucretius desires people to follow in the footsteps of Epicurus.