Here I describe a practical way to help imagine the 5th dimension & so on .
Imagine a sheet of paper. In this paper, you draw a line. This is your first dimension let's call it X, for illustration. Imagine that you can draw a point in this line, but in any position of this line, you can have only one point. You can't have more than one point in a specific position in this line. Your coordinate system for any point in this line (or axis) would be a number: say, (3).
Now you draw a line perpendicular to this one. Let's call it Y. Now, you can have points on the same X position, as long as they are in different Y positions. Your coordinate system could be represented as (x,y). So, now you can have as many points in X=3, as long as Y is different. If you think about the dimension X alone, you might see that (3,1) and (3,2) are in the same place, which would be impossible with one dimension only.
Let's go a bit further. Now you draw a line going OUT of the paper - say, going upwards with the paper standing on your desk. You can't do it with paper, because for all that matters we treat paper as a bi dimensional medium for drawing. But imagine a third Lind going out of the paper, anyway. Let's call this line Z. It's your third dimension. Now, your coordinate system can be represented as (x,y,z). If you look at the two first dimensions, you can have points in (3,2), which would be impossible with two dimensions only, but now it is, as long as the third dimension is different.
Three dimensions is how we perceive space. To draw more than this is difficult. But we can! Imagine your 3d paper. You can't have two points in the same space.... AT THE SAME TIME. So, how do you think about the fourth dimension? One can easily think of it as time. So, now you have (x,y,z,t). So, you can have multiple points on the same space, in different times.
Go one dimension further. I can't define any, but let's say it's called i (of incognito). Now, your coordinate system is (x,y,z,t,i). In this coordinate system, you can have points occupying the same poisition in space, at the same time, AS LONG AS THE POSITION IN i IS DIFFERENT. Some people propose this fifth dimension as alternative realities/universes.
Remember, this is not mathematical theory and I don't claim correctness. But it helps me visualize things for my work thinking in this incremental way (as the network of my supercomputer is a six-dimensional torus mesh).
Hope it helps ( Courtesy : Abhijeet Borkar, PhD in Physics (Astrophysics)
