Why have cities in the Western world grown so much in the last century?

Cities have grown so much because the more rural areas
can't, or at least aren't, sustaining their current populations. People have to move
where the jobs are.  Rural industries, such as agricultural pursuits, have shifted from
smaller, family-owned operations to conglomerations which require fewer workers and
certainly limit those who want to work for the love of the land.  Non-agricultural
enterprises need to be near population centers to find their work
forces.

Cities have also grown because of a lack of
services in less-populated areas.  Medical facilities and expertise can generally only
be found in larger population centers.  Wireless phone, cable, and internet services are
not always consistently available in the more remote rural areas.  Cultural
opportunities generally flourish in the city environment.  All of these are strictly
financial issues; services must be cost-effective to turn a profit, and the smaller
numbers prohibit cost-effectiveness outside of larger
cities.

Life in the city, while it certainly has its
trade-offs, is often more convenient to shopping and entertainment and transportation
and all the other things which take more time and effort in the more rural
places. 

I'm certain there are plenty of other reasons for
this pattern of migration, but this should help you get started.