What's the interpretation of setting small singular values to $0$ in SVD

Question

Given a linear transformation $A$, where $\dim(A) = m \times n$, we can decompose it using SVD: \begin{align} A = U\Sigma V^T \end{align} to a series of transformations, that is inverse rotation, scaling and rotation (transformation from sphere to ellipsoid). Interpretation of $A$ is obviously a transformation from $\mathbb{R}^n$ to $\mathbb{R}^m$. Let's assume that there are $k$ very small singular values in $\Sigma$. Although I feel that setting these $k$ values to $0$ is the right thing to do, I don't really know how it is justified in terms of reducing dimensionality. If semi-minor axis of an ellipse is short (because of the $\approx 0$ singular value), what does it really mean in terms of transformation, and what does it tell us about $A$?