Human intuition behind SVD in case of recommendation system

I checked the SVD for recommendation engine thread but it does not answer my question. I struggled very hard to understand the SVD from a linear-algebra point of view. But in some cases I failed to connect the dots. So, I started to see all the application of SVD. Like movie recommendation system, Google page ranking system, etc.

Now in the case of movie recommendation system, what I had as a mental picture is…

The SVD is a technique that falls under collaborative filtering. And what the SVD does is factor a big data matrix into two smaller matrix. And as an input to the SVD we give an incomplete data matrix. And SVD gives us a probable complete data matrix. Here, in the case of a movie recommendation system we try to predict ratings of users. Incomplete input data matrix means some users didn’t give ratings to certain movies. So the SVD will help to predict users’ ratings. I still don’t know how the SVD breaks down a large matrix to smaller pieces. I don’t how the SVD determines the dimensions of the smaller matrices.

It would be helpful if anyone could judge my understanding. And I will very much appreciate any resources which can help me to understand the SVD from scratch to its application to Netflix recommendation systems. Also for the Google Page ranking system or for other applications.

I am looking forward to seeing an explanation more from human-intuition level and from a linear-algebra point of view: how does the SVD work deep down from the core?