How do you know what a matrix represents in matrix decomposition?

Question

I am currently working on a project that involves matrix decomposition using SVD. I was wondering, how do people know what each decomposition represents? For instance, say the columns of a matrix represent various movies, and the rows represent users, and each $(x,y)$ pair represents a rating given by user $x$ to a movie $y$. Through SVD, one finds the decomposition: $$X = U\Sigma V^T$$ where $X$ represents the (user, movie) matrix, $U$ represents a user to latent characteristics matrix, and $V$ represents a movie to latent characteristic matrix. my question is, how do people know that's what $U$ and $V$ represent? Is there some proof?