Cause of skin effect: eddy current inside conductor
Yes, the magnetic field is densest at the surface, but the eddy currents are not.

The main current induces the magnetic field, H. The H field forms rings with the greatest intensity near the surface, and induces eddy currents. The eddy currents flow in rings around the magnetic field lines. It is impossible to make a ring around a field line on the very surface of the conductor, but more importantly, there is less space inside a ring-shaped field line than outside - the density of the eddy currents is greater inside a magnetic field "ring" than outside it. Furthermore, currents from more of these hypothetical rings overlap in the central part of the conductor. Thus, the eddy currents are strongest away from the surface. They do not cluster in the center as strongly - there is plenty even a short way from the surface, which is why the source current is reduced to traveling through the skin.

This image explains it:



In short, we have a series of reversals: Source current densest in the center induces a magnetic field strongest near the surface, which induces eddy currents stronger away from the surface, which cancel most of the source current, leaving most near the surface, even though the magnetic field is strongest there.