Why do we multiply matrices by column vectors? We get taught the procedure for carrying out such a task, but do you want to know what it represents?
In the following tutorial I use graphics animations to show you what's happening during the multiplication. The easiest place to start is by examining scaling multiplication, and how scaling matrices compare to the identity matrix.
Essentially, scaling matrices and identity matrices are the same thing, it's just that the values along the diagonal are either left as 1, which keeps the size the same, or set to some other value which then scales the vector (i.e. point in space).