I found this paper to be helpful. Hopefully you will find it useful too:
Paraphrasing what's explained in the article, if you understand SD, you can imagine that the squared deviation (x - mean of x)^2 can be represented graphically as a square (area = length of side ^2), where each side is the deviation from the mean for that observation. The numerator in the formula for variance is the sum of all n squared deviations, also known as SS, which graphically is the sum of the areas of all these squares. The sum of all these squares itself can also be represented graphically as a square, made up by smaller rectangles, all with the same width (square root of SS), but different heights (given by the squared deviation for that observation divided by the square root of SS). Again, the area of this bigger square is also SS. If you then convert this bigger square into (n-1) equal squares all with the same area, you get variance, which is the area of each one of these smaller square (aka SS/(n-1), or the formula for variance).
Interestingly, the length of each side of these variance squares will be equal to the standard deviation!