Interpretation of a simple SVD example

Question

I've recently started learning about SVD. This is a simple practical example in the book that I'm reading (Mathematics for Machine Learning):

This is a matrix (let it be $A$) which holds the rating (out of 5) for 4 movies -Star Wars, Blade Runner, Amelie, Delicatessen, given by 3 people -Ali, Beatrix, Chandra.

The SVD of this matrix is: $A=UEV^T$ =

Let the column vectors of $U$ be $u_1$,$u_2$,$u_4$,$u_4$. Let the column vectors of $V$ be $v_1$,$v_2$,$v_3$. The following assumptions made about this whole case:

1. All viewers rate movies consistently using the same linear mapping.
2. There are no errors or noise in the ratings.
3. We interpret the left-singular vectors $u_i$ as stereotypical movies and the right-singular vectors $v_j$ as stereotypical viewers.

Now the problem I'm having is, with the interpretation of this example.

In the book it says that the column's of $U$ and columns of $V$ can be thought of as different movies and movie watchers in their respective vector spaces. The elements can be thought of as values(amounts) of certain features that give the idea of what kind of movie or movie watcher it is. (If the columns of $U$ represent movies, how do we interpret the output of this mapping? Because the output is supposed to be a vector containing ratings for these movies by a user)

The past few days I read about this a bit. the interpretations about the columns of $U$ and $V$ still don't make sense to me, though the idea of extraction genres from a list of movies and their given ratings, through SVD is interesting., it still seems like just an interpretation.

Finding a matrix which is nearly the same by choosing what columns of $U$ and $V$ to use based on the non-zero values of $E$, essentially using this formula: $$ A = \sum_{i=1}^r\sigma_iu_i{v_i}^T $$ $$ A(k) = \sum_{i=1}^r\sigma_iu_i{v_i}^T $$ to find an approximation of the original matrix does make sense to me.