I don't see what you are asking. All that is said is that instead of calling the linear transformation "f", they are calling it "M". The second part of that sentence then says that we can always represent a linear transformation as multiplication by a matrix. Do you know what matrix multiplication is?

Presumably, it should be in this link:

If two vectors have the same direction but possibly different lengths the each is a numerical multiple of the other. If M "changes the magniude but not the direction" of m, then Mv= av where a is a number, then applying the linear transformation to this v is just like multiplying by a number, which is much easier.The rules for using a matrix to transform a vector are given in the article linear algebra.

If the action of a matrix on a (nonzero) vector changes its magnitude but not its direction, then the vector is called an eigenvector of that matrix.

Of course, it is true that M0= 0= a0 but if Mv= av for v non-zero, then we say that "a" is an "eigenvalue" of M and v is a corresponding "eigenvector"

It's hard to say more without knowing what youdoknow and understand about vectors, matrices, linear transformations, and vector spaces.