It's my understanding that any linear transformation can be represented as the multiplication of a vector by a matrix, resulting in another vector that generally (but not necessarily) is different from the "original" or "source" vector. There may be other ways to represent a linear transformation, but the matrix-vector representation is common.

As for your second question, the product of two matrices, provided the multiplication is defined, will yield another matrix. The new matrix can then be used to transform a vector, provided that the dimensions are right.

I hope this is clear.