Introduction
Note: This information is taken directly from VS Battles.
Summary
Higher dimensional manipulation is the ability to manipulate greater spatial and temporal dimensions than 3-D space, such as 4-D space and above.
According to infinity in projective geometry, and the concept of a Hausdorff dimension, each higher spatial (or added temporal) dimension is a more than countably infinite number of times greater than the preceding number.
An easy way to grasp this concept is as follows:
A 1-Dimensional (line) object only has length.
A 2-Dimensional (plane) object has length and width. The area of a 2-D object = length x width. The width of any 1-D object = 0, so its area = 0, even if its length = infinity.
This works in the same manner with 3-Dimensional space. The volume of a 3-D object = length x width x height. Since a 2-D object's height = 0, it doesn't matter if its length or width = infinity. Its volume, and mass, will still = 0.
"Hypervolume"/the 4-Dimensional volume analogue = length x weight x height x a fourth dimension. Since a 3-D object's fourth dimension = 0, its "hypervolume" and "hypermass" = 0
For a 5-Dimensional volume analogue = length x width x height x a fourth dimension x a fifth dimension. Since a 4-D object's fifth dimension = 0, its 5-D volume analogue, and 5-D mass analogue = 0
Basically, what this means is that, just like an entirely flat, two-dimensional square has a more than countably infinite number of times less volume (and mass) than a three-dimensional cube, the cube also has a more than countably infinite number of times less volume (and mass) than a four-dimensional tesseract, which has an infinite number of times less volume (and mass) than a five-dimensional hypercube, and so onwards.
Here is a story book that was among the first to take up the subject of different dimensions. Here is an episode of Futurama that does much the same thing. And here is a simple explanation of how even a 4-dimensional character relates to our 3-dimensional world.
Here is a Umineko explanation for how objects that appear infinite in lower-dimensional space, relate to ones in a higher dimension.