Why is multiplication between matrices defined this way?

The reason is a matter of convenience, and there is no proof to be found. The fact is that in many scientific disciplines it is often necessary to do a series of long and repetitive operations. The use of matrices simplifies the exposition of calculations and their reproducibility.

For example, imagine that you have a point P of coordinates (x, y, z). I want to rotate it around the x-axis first, then around the y-axis, and finally around the z one.

What are the new coordinates?

Remembering such a series of sums and multiplications is a nightmare, however I can at least formally solve the problem with 3 rotation matrices.

In each of the 3 matrices, I replace the argument of the sines and cosines with the value of the angle I need, then proceed to the calculation. For example, calling the 3 angles: α, β and γ, I can multiply the 3 matrices with each other and get a single matrix with the rotation values I am interested in.

At that point, I take the resulting matrix R, and I multiply it by my point P expressed as a column vector, and I get the new coordinates of the rotated point.

This is possible because the product between matrices was defined that way and produces the correct result. If it had been defined any other way…