I've recently come across this video where SVD is explained using so called concepts, on the example of users and film ratings. The matrix $U$, containing left singular vectors, is interpreted as "user-to-concept" similarity matrix, whereas matrix $V^T$ is interpreted as "movie-to-concept" similarity matrix.
Although this example seems somewhat intuitive and very illustrative, I have hard time understanding why we can think of SVD in such way. In terms of linear transformation SVD is simply a roration, followed by stretching and another rotation, but why are we so sure that the two first left singular vectors happen to be responsible for (in this case) sc-fi and romance concept, respectively?