I think this is a worthwhile question to ask, but I note that Maxwell had what I take to be *an *answer, that EM is "about" a system of vortices. I apologize if this seems facetious, but I take it that what is meant by "physical meaning", "intuitively", and "understood" partly depends on the period, the person, and the purpose.

For a pretty good "What is a Gauge" lecture, you could look at http://terrytao.wordpress.com/2008/09/27/what-is-a-gauge/, which I found by Googling for "gauge fixed electromag". [Thanks for the nice math.stackexchange link, BTW.]

Supposing we wanted to play a little, however, consider that in a classical context, if we have an electromagnetic potential $A_\mu(x)$ *and* a Dirac spinor field $U(x)$, we *could* note that (on Minkowski space or locally) there is a natural gauge fixing because $T^\mu(x)=\overline{U(x)}\gamma^\mu U(x)$ is a time-like 4-vector wherever $U(x)$ is non-zero, so we *could* fix a natural "Maxwell-Dirac axial gauge" by requiring that $T^\mu(x)\,A^{MD}_\mu(x)=0$. Given this fixing, and the mutually orthogonal and equal length 4-vectors $$Z^\mu(x)=\overline{U(x)}\gamma^\mu\gamma^5 U(x),\quad

X^\mu(x)=\Re[\overline{U(x)}\gamma^\mu U^c(x)],\quad

Y^\mu(x)=\Im[\overline{U(x)}\gamma^\mu U^c(x)],$$ which are orthogonal to $T^\mu(x)$, $T^\mu(x)Y_\mu(x)=T^\mu(x)Y_\mu(x)=T^\mu(x)Z_\mu(x)=0$, to each other, $X^\mu(x)Y_\mu(x)=Y^\mu(x)Z_\mu(x)=Z^\mu(x)X_\mu(x)=0$, and the same length up to a sign, $X^\mu(x)X_\mu(x)=Y^\mu(x)Y_\mu(x)=Z^\mu(x)Z_\mu(x)=-T^\mu(x)T_\mu(x)$, we *could* take the dynamics to be "about" a tetrad, a 4-vector $A_\mu^{MD}$, and a U(1) phase, $$(T^\mu(x), Z^\mu(x), X^\mu(x), Y^\mu(x), A^{MD}_\mu(x), \phi(x)).$$ $X^\mu(x)$, $Y^\mu(x)$, $Z^\mu(x)$, and $A^{MD}_\mu(x)$ are all space-like because they are orthogonal to $T^\mu(x)$, but $A^{MD}_\mu(x)$ is not otherwise constrained relative to the tetrad $(T^\mu(x), Z^\mu(x), X^\mu(x), Y^\mu(x))$. Note that fixing the U(1) gauge is related to fixing the arbitrary charge conjugation phase at every point.

$\phi(x)$ is the phase of the complex number $(\overline{U(x)}U(x), \overline{U(x)}\gamma^5 U(x))$, which *could* be taken to have one of a number of geometrical interpretations, but I've not found any approaches that are at all compelling (even just to me).

For *this* system, gauge invariance is just a way of writing down a particular dynamics for a system of 4-vectors that is nonlinearly constrained by various orthogonalities. We can play this game in various different ways, for example by using the bivector forms that can be constructed using $U(x)$ or by using the 3-form that can be constructed using $T^\mu(x)$ and the electromagnetic field, $T\wedge \mathrm{d}A$, but the trouble is that it's not clear when any such construction helps with either tractability or intuition, there's nothing unique about any given choice, and extending this kind of approach to a classical electroweak theory or to QED looks kinda bad. I also find the jump from a particular geometrical construction to a visual (or other sense) intuition not obvious.

Apologies that I haven't put this in a general Lorentzian manifold, but I don't play well in that context. Please also note that I've been thinking about whether this kind of construction might be useful for a while, so I took your question as an opportunity to write up one variant and see what it looks like.