Why does the thickness of a wire affect resistance?
The car analogy isn't such a good one, since electrons don't actually flow from one end of the wire to the other (well they do but extremely slowly) and it implies there is some space between the cars, whereas it would be more like a traffic jam whatever the width of the highway.
It's more like a line of billiard balls, and force is applied to the first one, and the energy is transferred to the last one through all the intermediate balls (a bit like newtons cradle, although the balls don't really bounce into each other). The free electrons bounce around, occasionally being impeded (see below) with the potential difference causing an average inclination to the direction of current.  

A water analogy is better - the pipe is always full of water, and for the same pump (battery), the pressure (voltage) is always lower the wider the pipe, which equates to more flow and a lower resistance.  

This quote from the Wiki page on resistivity explains reasonably well: 


  In metals - A metal consists of a lattice of atoms, each with an
  outer shell of electrons which freely dissociate from their parent
  atoms and travel through the lattice. This is also known as a positive
  ionic lattice.4
  This 'sea' of dissociable electrons allows the metal
  to conduct electric current. When an electrical potential difference
  (a voltage) is applied across the metal, the resulting electric field
  causes electrons to move from one end of the conductor to the other.
  Near room temperatures, metals have resistance. The primary cause of
  this resistance is the thermal motion of ions. This acts to scatter
  electrons (due to destructive interference of free electron waves on
  non-correlating potentials of ions)[citation needed]. Also
  contributing to resistance in metals with impurities are the resulting
  imperfections in the lattice. In pure metals this source is
  negligible[citation needed].
  The larger the cross-sectional area of
  the conductor, the more electrons per unit length are available to
  carry the current. As a result, the resistance is lower in larger
  cross-section conductors. The number of scattering events encountered
  by an electron passing through a material is proportional to the
  length of the conductor. The longer the conductor, therefore, the
  higher the resistance. Different materials also affect the
  resistance.