Intuitive explanation of Skin Effect?
If you want to understand this, instead of thinking of a transmission line as two wires with special behavior, you need to think of a transmission line as a guiding structure for electromagnetic waves. When we talk about currents and voltages and capacitance and inductance in a transmission line, that's a useful simplification for many cases. But to understand skin effect you should instead think about an electromagnetic wave that is guided by conductive structures. The currents and voltages that we measure on the transmission line are caused by the EM wave travelling along it, not the other way around.

So then, when an EM wave encounters a conductive material, what happens? The EM wave is reflected. It doesn't penetrate deeply into the conductor. But if the conductivity is not infinite, it does penetrate slightly. And up to the depth that it penetrates, it causes currents to flow in the conductor. That is the skin effect.


  I'm slightly confused as to the co-ordinates of how you're defining your transmission line.


Normally we call the axial direction (along the length of the transmission line) the z-dimension. The transverse dimensions are x and y.

It's easiest to visualize in a parallel plate waveguide. There, the propagating mode can be seen as a sum of plane waves reflecting back and forth between the two plates. So these plane waves have a k-vector that has components in both the z and x directions. The reflections off the conductors is what causes the currents in the conductors, and the depth of penetration into the conductors is what determines the skin depth.