This has nothing to do with it. Field lines work because the source-free Maxwell equations imply that the 2-form F is harmonic
Harmonic forms (in any dimension) have the distinction that they capture purely topological information. If you integrate a harmonic n-form over a closed n-surface, the result is either zero or non-zero, depending on whether the n-surface encloses some topological feature (for example, a 1-surface on a cylinder might wrap around the cylinder...or a 2-surface in R^3 might enclose a charge). Any
n-surface that encloses the same set of topological features must give you the same result.
You can think of this as a higher-dimensional analogue of contour integration. In fact, all analytic functions on the complex plane satisfy Laplace's equation, which is why contour integration works.