Interpretation of an SVD for recommender systems
First of all, note that the dot product between two movies/users is by definition the correlation between them. So your intuition for treating it as a similarity measure is not wrong.

Now, applying SVD to $A$ is simply a re-representation of it in some other basis. Let's call it feature space. If we'll take it one step further and keep only $k$ leading vectors of $U$ and $V$, we'll get two matrices: $U^{(k)} \in \mathbb{R}^{f,k}$ and $V^{(k)} \in \mathbb{R}^{k,u}$ (where $f$ and $u$ are the dimensions you supplied in your question). where $U^{(k)}\cdot V^{(k)} \approx A^{(k)}$ is a $k$-rank approximation for $A$ ** (note it has the same dimensions).

So you can of $U^{(k)}$ as a matrix which rows movies, and each movie has some $k$ latent features. Similarly, $V^{(k)}$ will be a "list" (columns) of users and each user is described by some $k$ values (latent features). 

Note that you can "split" the scaling matrix $\Sigma^{(k)}$ between $U^{(k)}$ and $V^{(k)}$: $U'^{(k)} = U^{(k)}\Sigma^{(k)0.5}$ and $V'^{(k)} = V^{(k)}\Sigma^{(k)0.5}$, to make up for the lack of scale of $U$ and $V$.

This factorization can be elaborated into more interpretable methods such as LDA.

** The variational approach to singular values tell us that this approximation minimizes the difference between $A$ and any other $k$ rank matrix in terms of Frobenius-norm and operator-norm (and possibly other norms I'm not aware of).