It's of course a metaphor to say ##\vec{E} \cdot \mathrm{d}^2 \vec{f}## is a "flow through the surface element". For a static electric field nothing is really flowing in the literal sense.
The metaphor of course has its origin from some real flow. The most simple case is that you have some conserved quantity flowing (in the literal sense) with the matter. In electromagnetism you have this case concerning the electric charge, which is a conserved quantity carried with the electrically charged matter, on a microscopic level it's carried by the atomic nuclei (positive charge) and electrons (negative charge), but in classical electrodynamics as in all of classical physics we don't consider this microscopic resolution at the atomic level but take averages over macroscopically small but microscopically large regions of space.
Then you can describe the charge in a macrocsopic continuum mechanical way. Here you describe it by the charge density ##\rho(t,\vec{x})##. The meaning is that if you measure at time ##t## the charge in a small volume element ##\mathrm{d}^3 x## around the position ##\vec{x}## the (net) charge contained in this volume element is ##\mathrm{d} Q=\mathrm{d}^3 x \rho(t,\vec{x})##.
Now we also know that charge is conserved, which means that (in this macroscopic picture) the charge contained in the volume element ##\mathrm{d}^3 x## can only change, because there is some charge flowing through the boundary of this volume, which is a closed surface. Let now ##\vec{v}(t,\vec{x})## be the flow field of the charged medium, i.e., if you look at time ##t## at the position ##\vec{x}## the matter just being there has the velocity ##\vec{v}(t,\vec{x})##.
Now we consider the charge flowing through the surface. For that purpose at any point ##\vec{x}## on the surface you define the surface normal element ##\mathrm{d}^2 \vec{f}##, which is a vector being directed perpendicular to the surface pointing out of the volume (by convention) with the magnitude being the area of this surface element. Then in an infinitesimal time ##\mathrm{d} t## the particles just at the surface element going out of the volume is given by ##\mathrm{d}^2 \vec{f} \cdot \vec{v}(t,\vec{x})##. By choice of the direction of ##\mathrm{d}^2 \vec{f}## if the particles go out of the volume that's counting as positive, and if the particles go into the volume it's counted negative. So the particles sweep out a volume of ##\mathrm{d} t \vec{v} \cdot \mathrm{d}^2 \vec{f}##, and the charge being carried through the surface is given by $$\mathrm{d} Q=\mathrm{d} t \mathrm{d}^2 \vec{f} \cdot \vec{v}(t,\vec{x} \rho(t,\vec{x}).$$
Now we can make the charge balance to express the charge-conservation law. The change of the charge contained in the volume during the infinitesimal time interval ##\mathrm{d} t## is
$$\mathrm{d} Q_V=\mathrm{d} t \partial_t \rho(t,\vec{x}) \mathrm{d}^3 x).$$
On the other hand we know that this change can only be due to charge flowing through the surface. Now with our convention of directing the surface elements ##\mathrm{d}^2 \vec{f}## out of the volume this means
$$\mathrm{d} Q_V = -\mathrm{d} Q=-\mathrm{d} t \mathrm{d}^2 \vec{f} \cdot \vec{v}(t,\vec{x}) \rho(t,\vec{x})$$
and thus you get
$$\partial_t \rho(t,\vec{x}) \mathrm{d}^3 \vec{x} = -\mathrm{d}^2 \vec{f} \cdot \vec{v}(t,\vec{x}) \rho(t,\vec{x})$$
or integrating over a finite volume ##V## with its boundary ##\partial V## you get
$$\dot{Q}_V=\int_V \mathrm{d}^3 x \partial_t \rho(t,\vec{x})=-\int_{\partial V} \mathrm{d}^2 \vec{f} \cdot \vec{v}(t,\vec{x}) \rho(t,\vec{x}).$$
Usually you introduce the current density,
$$\vec{j}(t,\vec{x})=\rho(t,\vec{x}) \vec{v}(t,\vec{x})$$
and write
$$\dot{Q}_V=\int_V \mathrm{d}^3 x \partial_t \rho(t,\vec{x})=-\int_{\partial V} \mathrm{d}^2 \vec{f} \cdot \vec{j}(t,\vec{x}).$$
This shows that the current density is a vector field telling you how much charge is flowing through a surface element ##\mathrm{d}^2 \vec{f}##, ##\mathrm{d} Q=\mathrm{d}^2 \vec{f} \cdot \vec{j}(t,\vec{x})##.
An important mathematical theorem now is Gauss's Theorem, according to which
$$\int_V \mathrm{d}^3 x \vec{\nabla} \cdot \vec{j}=\int_{\partial V} \mathrm{d}^2 \vec{f} \cdot \vec{j},$$
where the surface normal elements of the closed boundary surface ##\partial V## of the volume ##V## are defined to be pointing outward of the volume. Here the divergence of the current (in Cartesian coordinates) is
$$\vec{\nabla} \cdot \vec{j}=\frac{\partial j_1}{\partial x_1} + \frac{\partial j_2}{\partial x_2} +\frac{\partial j_3}{\partial x_3},$$
and the above derived charge-conservation law reads
$$\int_V \mathrm{d}^3 x \partial_t \rho(t,\vec{x})=-\int_V \mathrm{d}^3 x \vec{\nabla} \cdot \vec{j}(t,\vec{x}).$$
Now you can make the volume infinitesimally small again and then consider the fields you integrate over as constant within this small region. Then the volume element ##\mathrm{d}^3 x## cancels on both sides and you get the local form of the charge-conservation law,
$$\partial_t \rho(t,\vec{x})=-\vec{\nabla} \cdot \vec{j}.$$
Now back to the original idea to calculate the "flow" of the electric field through a surface. This comes from taking the volume integral
$$\int_V \mathrm{d}^3 x \vec{\nabla} \cdot \vec{E}=\int_{\partial V} \mathrm{d}^2 \vec{f} \cdot \vec{E},$$
where we have used again Gauss's integral theorem. Using Coulomb's law you can derive that
$$\vec{\nabla} \cdot \vec{E}=\frac{1}{\epsilon_0} \rho.$$
This is named Gauss's Law for the electric field. The factor ##1/\epsilon_0## is due to our choice of SI units with the Coulomb as the unit of the electric charge.
Using this in the above integral you get
$$Q_V=\int_V \mathrm{d}^3 x \rho(t,\vec{x})=\epsilon_0 \int_{\partial V} \mathrm{d}^2 \vec{f} \cdot \vec{E},$$
i.e., (up to the unit-conversion factor ##\epsilon_0##) the "flow of the electric field" through any closed boundary surface ##\partial V## of the volume ##V## gives the amount of charge contained in this volume.
It's clear that "flow" is here to be understood in the sense of being a mathematical analog to the ideas about something really flowing like the electric charge considered above.
