Some concepts in mathematics like Hindu-Arabic numerals, imaginary numbers, vectors and matrices, exist primarily for convenient manipulation and communication of concepts that represent quantity, structure, space and change. Such convenience is merely a matter of taste. It is perfectly possible to do all of modern mathematics using Roman numerals. Roman numerals can even replace matrices. Unfortunately in such a case mathematics becomes unnecessarily complex, and it becomes hard to be creative, recognize patterns and relations when tied up with such a complex set of rules.
As for exactly what matrices are for. You need to know what vectors are, and what they are for. To put it in English, vectors are anything which 1. when combined with another vector results in a vector. 2. when combined with itself any number of times also results in a vector. For example, a cocktail can be thought of as vector. Wait no. I was wrong: if a cocktail was a vector then the individual ingredients like vodka also ought to be cocktails, because they are what must be combined to form cocktails. Also "nothing" ought to also be a cocktail too if cocktails were vectors.
Now if cocktails were vectors (unfortunately they are not because of reasons above), then a matrix is like a recipe which if performed on a single cocktail can transform it to another cocktail.
That is most non-mathematical way I can describe a matrix.