What was progressivism, its religious and intellectual roots, and why were so many people drawn to it?

Progressivism is all about making progress -- and by that we mean social progress.  Progressives are aiming at making society better, fixing things that are wrong, reforming programs or patterns so that people's lives are improved.  
On the intellectual side, the roots of Progressivism are in the Enlightenment. Beginning in the 18th century, social theorists put forth many ideas on how society could be run more rationally and fairly for the benefit of all citizens.

Typically, religious folks were on the opposite side of these "liberal" ideas that stemmed from the Enlightenment.  But as the social gospel movement gained momentum, that changed.  The social gospel emphasized that godly people should be helping their neighbors in this world, not just preparing for a life in heaven after this world.

So, with both non-religious backing (in Enlightenment ideas) and religious backing (in social gospel ideas), the Progressive movement was poised to draw a lot of followers and supporters.