Action of $GL_n(k)$ on the space of polynomials in $n^2$ indeterminates

Question

I'm having a hard time with a proof where the author didn't specify the action. He just wrote: "Note that $GL_n(k)$ acts on $k\big[X_{1,1},...,X_{n,n}\big]$ by substitution and leaves $k\big[X_{1,1},...,X_{n,n}\big]_{\leq N}$ invariant".

I really don't understand what the author meant there.