I think, since $K$ is algebraically closed, all maximal ideals are of the form $m=(X_1-a_1,\dots,X_n-a_n)$. To specify a ring homomorphism from $K[X_1,\dots,X_n]$ to any other (commutative) ring $A$, you only need to specify a ring homomorphism $\rho:K\rightarrow A$ and $n$ points in $A$ that the $X_i$ will be mapped to. So in your case you can take the following ring homomorphism : $\rho:K\hookrightarrow K[X_1,\dots,X_n]$ and send $X_i$ to $X_i+a_i$. This defines a unique ring homomorphism $K[X_1,\dots,X_n]\rightarrow K[X_1,\dots,X_n]$, and it's an isomorphism (you can construct its inverse in the same manner). This sends your maximal ideal $m=(X_1-a_1,\dots,X_n-a_n)$ to $(X_1,\dots,X_n)$.