The "dream" does not "come true". Bruce's character, Cole, has a memory from his childhood where he sees a man get shot in an airport. This experience is traumatic, and gives him an uncommon strength of mind. This experience was totally real, and Cole remembers it.
This strength of mind, and his status as a criminal (which I don't remember being explained) make him a candidate for the time travel experiment. Twelve Monkeys style time travel is apparently rough on your mind. We see Cole struggling to remember his instructions (finding a spider, and eating it?), and the names of Cities. Not everyone is up to it, but Cole is.
Cole ends up going back in time, one more time, because he's in love, winds up at the airport, and we see those events that he remembers, but this time with Cole as an adult.
It's a mild predestination paradox. Cole can be a time traveler because he saw himself get killed, and he only got killed because he was a time traveler.
The movie was inspired by the French Film La Jetée, discussed in this question, in which a narrator explains most of this briefly.
As for whether the scientist/"insurance" agent lady is there to prevent the plague, get a sample of the disease, or is there by total accident, there is a lot of debate, like here.
