I'll describe a simple gauge theory geometrically, and then show how $A$ and $F$ relate to it.

Gauge theories describe the connectivity of a space with small, symmetric extra dimensions

Start with an infinite cylinder (the direct product of a line and a small circle). The cylinder can be twisted. To avoid appealing to concepts that I'm trying to explain, I'll just say that the cylinder is made of wire mesh: evenly spaced circles soldered to wires running the length of it. The long wires can rotate as a unit, introducing an angular twist between each pair of adjacent circles. It's clear that any such configuration can be continuously deformed into any other: all such cylinders are equivalent from the perspective of the proverbial ant crawling on them.

Replace the line with a closed loop, so that the product is a torus (and think of the torus as a mesh doughnut, even though varying the plane of the small circles like that technically breaks the analogy). Any portion of the doughnut short of the whole thing can be deformed into the same portion of any other doughnut, but the doughnuts as a whole sometimes can't be, because the net twist around the doughnut can't be altered. The classes of equivalent doughnuts are completely characterized by this net twist, which is inherently nonlocal.

Replace the loop (not the small circle) with a manifold of two or more dimensions. It's true, though not obvious, that the physical part of the connection is completely given by the integrated twist around all closed loops (Wilson loops).

$A$ and $F$ quantify the connectivity

In the discrete case, the connection can be described most simply by giving the twist between adjacent circles. In the continuum limit, this becomes a "twist gradient" at each circle. This is $A_\mu$, the so-called vector potential.

Any continuous deformation can be described by a scalar field $\phi$ representing the amount that each circle is twisted (relative to wherever it was before). This alters $A_\mu$ by the gradient of $\phi$, but doesn't change any physical quantity (loop integral).

The description in terms of Wilson loops, $\oint_\gamma A \cdot \, \mathrm dx$, is more elegant because it includes only physically meaningful quantities, but it's nonlocal and highly redundant. If the space is simply connected, you can avoid the redundancy and nonlocality by specifying the twist only around differential loops, since larger loops can be built from them. The so-called field tensor, $\partial_\nu A_\mu - \partial_\mu A_\nu = F_{\mu\nu}$, gives you exactly that.

(If the space is not simply connected, you can still get away with the differential loops plus one net twist for each element of a generating set of the fundamental group. The torus was of course a simple example of this.)

The force comes from the Aharonov–Bohm effect

Consider a scalar field defined over the entire space (unlike the earlier fields, this one takes a value at each point on each circle). The field is zero everywhere except for two narrow beams which diverge from a point and reconverge somewhere else. (Maybe they're reflected by mirrors; maybe the space is positively curved; it doesn't matter.)

Unless the field is constant across the circles, the interference behavior of the beams will depend on the difference in the twist along the two paths. This difference is just the integral around the closed loop formed by the paths.

This is the (generalized) Aharonov–Bohm effect. If you restrict it to differentially differing paths and use $F_{\mu\nu}$ to calculate the effect on the interference, you get the electromagnetic force law.

You can decompose the field into Fourier components. The Fourier spectrum is discrete in the small dimension. The zeroth (constant) harmonic is not affected by the twisting. The second harmonic is affected twice as much as the first. These are the electric charges.

In reality, for unknown reasons, only certain extra-dimensional harmonics seem to exist. If only the first harmonic exists, there's an equivalent description of the field as a single complex amplitude+phase at each point of the large dimensions. The phase is relative to an arbitrary local zero point which is also used by the vector potential. When you compare the phase to the phase at a nearby point, and there is a vector-potential twist of $\mathrm d\theta$ between them, you need to adjust the field value by $i \, \mathrm d\theta$. This is the origin of the gauge covariant derivative.

Circles generalize to other shapes

If you replace the circles with 2-spheres, you get an $\mathrm{SU}(2)$ gauge theory. It is nastier numerically: the symmetry group is noncommutative, so you have to bring in the machinery of Lie algebra. Geometrically, though, nothing much has changed. The connectivity is still described by a net twist around loops.

One unfortunate difference is that the description of charge as extra-dimensional harmonics doesn't quite work any more. Spherical harmonics give you only the integer-spin representations, and all known particles are in the spin-0 or spin-½ representations of the standard model $\mathrm{SU}(2)$, so the particles that are affected by the $\mathrm{SU}(2)$ force at all can't be described this way. There may be a way to work around this problem with a more exotic type of field.

I have nothing insightful to say about the $\mathrm{SU}(3)$ part of the Standard Model gauge group except to point out that the whole SM gauge group can be embedded in $\mathrm{Spin}(10)$, and I think it's easier to visualize a 9-sphere than a shape with $\mathrm{SU}(3)$ symmetry.

General relativity is similar

In general relativity, the Riemann curvature tensor is analogous to the field tensor; it represents the angular rotation of a vector transported around a differential loop. The Aharonov-Bohm effect is analogous to the angular deficit around a cosmic string. Kaluza-Klein theory originally referred to a specific way of getting electromagnetism from general relativity in five dimensions; now it often refers to the broad idea that the Standard Model gauge forces and general relativity are likely to be different aspects of the same thing.