When Spencer was doing an experiment on Zack for his sleeping habits for the science fair, Spence discovers that Zack can enter other people's dreams. Zack fell asleep after Vernon did, so Zack invaded Vernon's dream.
The case is that Zack invade other people's dreams when both are sleeping near each other, and as a result, Zack does not sleep well. Zack enters Gwen's dream for Spencer's experiment for the Science Fair and to get tickets for the concert for Cam, Gwen thinks that Zack is "the man of her dreams" so the 3 kids have to finds away to help with Zack's problem of lack of sleep.