A matrix tells you where the two basis arrows go. Every other point follows by linearity.
The columns of the matrix are the images of the standard basis vectors. An arbitrary vector is a linear combination of these basis vectors, so its image uses the same coefficients. The animation linearly interpolates from the identity to A; its intermediate maps may be singular.
Matrix multiplication moves vectors, not just their endpoints independently of the origin.
The colored parallelogram is the image of the unit square. Matrix entries are the final transformation A.