Perhaps the earliest novel-length treatment of alternate dimensions is William Hope Hodgon's The House on the Borderland (1907), in which the hero finds himself in a house that contains a portal to another dimension in which time moves at an accelerated rate.
Extension into other dimensions are featured in Miles J. Breuer's "The Captured Cross-Section" (1929), Donald Wandrei's "The Monster from Nowhere" (1935), Robert Heinlein's "And He Built a Crooked House" (1941), Martin Gardner's "No-Side Professor" (1946), Arthur C. Clarke's "Wall of Darkness" (1949), Bruce Elliott's "The Last Magician" (1951), and David Duncan's Occam's Razor (1957).
In their 1941 short story "The Street That Wasn't There," Clifford D. Simak and Carl Jacobi explain that "in modern astrophysics and mathematics we gain an insight into the possibility and probability that there are other dimensions, other brackets of time and space impinging on the one we occupy."
The premise of Simak's story is that our world exists, unknown to us, in a dimension created collectively by the combined minds of all humans in existence. When a plague wipes out a large portion of the population, the dimension-creating power of humans as a species is diminished, until our reality gradually fades away.
Another explanation for multiple dimensions is that alternate realities are created ever time someone makes a decision. For instance, if you decide to skip breakfast today, the decision creates two parallel dimensions: one in which you ate breakfast and another in which you skipped it. The universe in this way continually splits into parallel time tracks, a notion suggested by Philip Jose Farmer in his 1952 short story "Sail On, Sail On."
"String theory," a hypothesis that attempts to unify general relativity with quantum mechanics, says that the universe is made up of incredibly small strings vibrating in a space-time continuum consisting of 11 dimensions - the major dimensions of length, width, height, and time, plus seven "minor" (smaller) dimensions.
We live in a four-dimensional world, so we cannot see the other 7 dimensions. The strings vibrate in our four dimensions as well as the other seven dimensions, providing a connection between our dimension and others, and giving rise to the possibility of a parallel universe existing in those other dimensions.