How does minimizing MSE maximize variance? I was reading about PCA and they had this interesting line that "A line or plane that is the least squares approximation of a set of data points makes the variance of the coordinates on the line or plane as large as possible." Which I think makes sense visually if you looked at a scatter plot in 2D space of points and trying to project a line through them, but I can't think of a proof or more logical explanation for why this would be true.

Let's assume we've centered the data at zero.  For any point $x$ in $p$ dimensions you can write the squared distance to zero, $\|x-0\|_2^2$, as the squared distance in the projected two dimensions plus the (residual) squared distance at right angles to the page. That's Pythagoras' theorem.
Averaging over all the points, the mean squared distance to the points in the original $p$-dimensional space (which is fixed at whatever it is) is the sum of the mean squared distance in the two-dimensional projection and the mean squared residual error  of the projection.  Since the sum is fixed, when you minimise the second of these, you maximise the first.
