To add to what @Sycorax commented:
If you're satisfied with merely being a consumer of scientific
software, you can skate by without much knowledge at all. But if
you're curious and interested in really understanding what's going on
when your script runs, knowledge of statistics, linear algebra,
calculus and numerical optimization are essential; the more, the
So specifically linear algebra, what do you need to be a better user, understand software (and methods) better, and for modeling (but not being interested, at this moment at least, in writing software or proving theorems). The linear model is the backbone of much of statistics, so you need to understand that well. Since a linear model can be seen geometrically as a subspace of a certain linear space, and this is just the basics of linear algebra: spaces, subspaces, linear combinations, linear independence (and dependence). You need this to understand the specification of a linear model, and the various problems that can occur.
Then the basics of matrices, linear equations in matrix form, solutions, matrix inverses. And a lot of linear algebra this days in embedded in matrix factorizations, the spectral decomposition, the svd (see What is the intuition behind SVD?), the QR decomposition (see Understanding QR Decomposition.)
I found it useful to get a better intuition for matrix multiplication, see https://math.stackexchange.com/questions/198257/intuition-for-the-product-of-vector-and-matrices-xtax/198280#198280.
So where to learn this? Assuming you have learnt a little linalg in the past, I would start with some linear models book with an appendix an matrices. For instance Linear Regression Analysis (2nd Edition) by Seber and Lee but there are no lack of competition. And read Venables & Ripley MASS (the book, fourth edition) section 6.2 Model Formulae and Model Matrices.