Suppose $X$ and $Y$ are discrete random variables such that
$$Y = aX+b$$
where $a$ and $b$ are scalars. It is easy to find a geometric interpretation of the following two facts
$$\Bbb E[Y] = a \,\Bbb E[X] + b \tag{1}$$
$$\mbox{Var}[Y] = a^2 \, \mbox{Var}(X) \tag{2}$$
$(1)$ can be interpreted as follows. If we move the (values of random variables) points (on the number line) in a linear fashion then the mean of those points also will change in same linear fashion and one can arrive to the geometric interpretation of $(2)$ also.
Is there any geometric interpretation to understand the following?
$$\mbox{Var}(X) = \Bbb E \left[X^2 \right] - \left( \Bbb E[X] \right)^2$$
I think a trivial interpretation, but maybe it works for you. The variance is a measure (MSE) of how far the points are from the mean. If one moves the atoms in a linear fashion, the bias should not impact the variance but the scale does.