Why is wire's inductance inversely proportional to its thickness?
First consider two straight parallel wires separated by some distance, with current running in the same direction. The total inductance of these two wires (consider the current to be the sum of the current in each) is the sum of the self inductance of each wire, plus the mutual inductance between the two. Because the currents are running in the same direction, this mutual inductance adds as a positive contribution to the total inductance (i.e. a positive coupling constant). The mutual inductance of these two wires gets larger if the wires are moved closer together, and therefore the total self inductance of the two-wire 'system' also increases.

Next, think of the current in a single conductor as a bunch of small, individual current elements at their various locations within the cross section. The total inductance is the self inductance of each small current element, plus the sum over all the mutual inductances between every pair of current elements. If the wire is smaller in diameter, these current elements are all closer to one another, and so the mutual inductance increases, therefore increasing the total inductance of the conductor.

You could also approach the argument from the standpoint of the energy stored in the magnetic field, but I think this is more intuitive.