John Schwarz and Michael Green set out in 1984 to discover the laws of quantum gravity and made a breakthrough. This breakthrough only worked if the "bulk" had 9 space dimensions and 1 time dimension, six more than space dimensions than our "brane." In this "superstring theory" the extra dimensions have large influences on our "brane," that can be measured if we had sufficient technology. These influences could make it possible to reconcile the laws of quantum physics with Einstein's laws of relativity.
In Interstellar the bulk's six extra dimensions are made into one extra dimension for practice purposes. The bulk in interstellar shares 3 space dimensions and 1 time dimension with our brane. It also has a fifth space dimension. Kipp Thorne refers to this as "out back," and extends perpendicular to our universe.