Explain why cities have grown so explosively in the western world (primarily Europe and North America) during the last century and a half or so...

I think that there is more that has to be taken into
account when we look the phenomenon of urbanization.  We cannot explain urbanization
without mentioning the increase in farm productivity and the advances in transportation
that allow excess farm produce to be moved to
cities.

Before mid-1850s, it would have been hard for
cities to grow very large.  Farms were not productive enough to support huge urban
populations and many people were needed in the rural areas to work on the farms.  As
mechanization and other technologies increased farm productivity, farms came to be able
to feed larger urban populations and there was less need for people to live in rural
areas and work on farms.

In addition, advances in
transportation (notably the railroads) allowed urbanization to proceed.  If it had not
been for better transportation, the huge surpluses produced by farms in places like the
Midwest US could not have been used to feed the huge urban populations of the East
Coast.

Thus, we have to take into account the increasing
ability of small farm populations to feed huge urban populations.  This made
urbanization possible.