Relevance / Importance of the category Mat
In my opinion, the two most significant things to learn from the example are:
Morphisms are important
One thing I really like about Mat is that it's a familiar example where our prior experience is to consider all of the importance to be in the morphisms, which serves to contrast with other examples like AbGrp where prior experience tends to be object-centric.
A running theme throughout category theory is the emphasis on the importance of morphisms — a point of view that can even be fruitfully taken to the extreme of considering objects to be wholly irrelevant beyond their role of being the sources and targets of morphisms.
E.g. in many of the familiar examples, all of the usual structure of the objects can be recovered from morphisms. For example, in Top, given a topological space $X$ one can identify its set of points with $\hom(1, X)$ and its family of open sets with $\hom(X, 2)$, where $1$ is the one-point space and $2$ is the Sierpinski space.
Categories are natural structures
Another thing I like about Mat is it demonstrates a new way in which categories naturally organize familiar structure.
Matrix algebra is sort of an oddball in abstract algebra because the product is only partially defined; it doesn't really fit well with the usual approaches to the subject; you have to do unnatural things like restrict your attention only to square matrices of fixed dimension, or do weird things like allow all products but define those with mismatch to multiply to zero.
But lo and behold, the structure of a category happens to be exactly how matrices with the matrix product want to be organized.
Another point about the example is it paves the way of applying ideas of category theory to linear algebra.
It wasn't until I learned about Mat, for example, that I really accepted that "column space of an $n \times m$ matrix" is a better notion for computation than "subspace of $\mathbb{R}^n$". Or finally allowed "$n \times 1$ matrix" to replace "element of $\mathbb{R}^n$" in my mind when doing calculations.
Mat is basically the same category as Vect of finite dimensional with a distinguished basis. It's to demonstrate the notion of categorical equivalence as well as a cute example of a category whose morphisms are not functions.
Is that category really so trivial? I think part of the point is that while the objects are rather "boring", there is much more richness to be found in the morphisms.