Let $x \in M_n(F)$. If the characteristic polynomial of $x$ has distinct prime factors in the ring $F[t]$, then there is some idempotent that commutes with everything that commutes with $x$, hence, if the centralizer is maximal, then it is the centralizer of that idempotent. The centralizer of an idempotent is clearly maximal. So this gives one kind of maximal subgroup.

In the remaining kind, the characteristic polynomial has the form $f(t)^k$ for some irreducible polynomial $f(t)$. Here we split into two cases - either $x$ is semisimple or it isn't. If $x$ is semisimple, its centralizer is a representation $M_k(F')$ where $F'$ is an extension of $F$. The elements which commute with this ring are just the elements of $F'$, and the only ones that have a larger centralizer are the subextensions. So this algebra is maximal if and only if $F'$ is an extension of prime degree. This gives a second kind of maximal subgroup.

If $x$ is not semisimple, we can take the Jordan decomposition $x=x_{ss}+x_n$, and the centralizer of $x$ is contained in the centralizer of $x_n$, but $x_n$ is not in the center of $M_n(k)$, so we may assume $x=x_n$. Then by taking a power we may assume $x^2=0$. In this case, we can check by looking explicitly at the Jordan block that anything that commutes with the centralizer of $x$ must be a linear combination of $1$ and $x$, and so the centralizer of $x$ is maximal.

So there are three cases:

The algebra is the centralizer of an idempotent, the product of two matrix algebras, as in Amitanshu's example.

The algebra is the algebra of matrices over a prime-degree extension of $F$.

The algebra is the centralizer of a nilpotent $x$ satsifying $x^2=0$.

propersubrings (otherwise $M_n(F)$ is the unique maximal centralizer). $\endgroup$