One way to write Maxwell's equations is
\begin{align*}
\partial_a F^{ab} &\propto J^b
\tag{1a}
\\
F_{ab} &=\partial_a A_b-\partial_b A_a.
\tag{1b}
\end{align*}
I'm using the usual summation convention for repeated indices.
I'm assuming flat spacetime with the usual Minkowski metric $\eta_{ab}$, and I'm using this metric to raise and lower indices. Equation (1b) says that the EM field bivector $F_{ab}=-F_{ba}$ can be written in terms of a gauge field $A_a$ as shown, and then equation (1a) may be regarded as the equation of motion for $A_a$ with the given current $J^b$.
If we don't want to use the gauge field $A_a$, we can also write Maxwell's equations like this:
\begin{align*}
\partial_a F^{ab} &\propto J^b
\tag{2a}
\\
\partial_a F_{bc}+
\partial_b F_{ca}+
\partial_c F_{ab} &= 0.
\tag{2b}
\end{align*}
Equation (1b) is a solution of equation (2b). If the spacetime is topologically trivial, then this is the most general solution. (If we want to allow non-trivial topology, then equation (2b) is more general than (1b).)
Equation (2b) is completely antisymmetric in all three subscripts. This fact is important in the next way of writing the equations.
Geometric Algebra is another name for what mathematicians and physicists usually call Clifford Algebra. Dirac matrices are a matrix representation of the Clifford algebra associated with Minkowski spacetime. This Clifford algebra is an associative algebra in which the basis vectors $\gamma^a$ satisfy
$$
\gamma^a\gamma^b + \gamma^b\gamma^a = 2\eta^{ab}.
\tag{3}
$$
(I'm using the conventional notation for Dirac matrices because it's familiar, but we don't really need any matrix representation here. We only need the associative algebra.)
Using this algebra, Maxwells equations (2a)-(2b) may be written as a single equation like this:
$$
\partial F\propto J
\tag{4}
$$
with
$$
F = \gamma^a\gamma^b F_{ab}
\hskip2cm
J=\gamma^a J_a
\hskip2cm
\partial=\gamma^a\partial_a.
\tag{5}
$$
When equation (5) is expanded in the given basis, it has two parts:
A vector part, which is a linear combination of individual $\gamma^a$s. This part gives equation (2a).
A trivector part, which is a linear combination of products $\gamma^a\gamma^b\gamma^c$ with all three indices distinct from each other. This part gives equation (2b).
In my experience, for most purposes, equations (1) or (2) are easier to use than equation (4). However, equation (4) does have a few nice uses. Here's one example: if $J=0$, then we can apply $\partial$ to equation (4) on the left to get $\partial\partial F = 0$, and we can use associativity combined with the identity $xx = x_a x^a$ to get the wave equation $\partial_a\partial^a F_{bc} = 0$. This isn't much of an advantage (if any), because the wave equation can also be derived just as easily by contracting equation (2b) with $\partial^a$ and then using the $J=0$ version of equation (2a). The best use for the Clifford-algebra formulation might be for studying how $F$ transforms under Lorentz transformations. The basic idea is described in another post. (That post is written for Euclidean signature instead of Lorentzian signature, but the idea is the same.)
