> Why would we ever want a basis besides the standard basis in real Euclidean space?

It think it can be explained from a SVD pointview, the standard basis may not likely be optimal one, and didn't capture the most variance. The basis with larger variance, will presumably contains more information.

After obtaining the good basis, we can approximate the original matrix using only the ones(often much smaller in number than standard basis) with larger variances. This is useful to find a shorter representation, yet preserves much of the distance information.

I think this post explain the intuition pretty clear:

http://www.cs.princeton.edu/picasso/mats/PCA-Tutorial-Intuit...