How did the ideas behind Renaissance architecture come into play, and where did the parts come from? I know Renaissance columns come from Rome, but...

Like many other features of society, architecture changed dramatically during the Renaissance. The initial impetus seems to have been the arrival of ancient Greek and Roman texts from the Middle East, which had been preserved and studied by Middle Eastern scholars but largely forgotten in Europe.

Two major forces influenced just about everything in the Renaissance:
(1) Trying to emulate ancient Greece and Rome
(2) Trying to invent and discover fundamentally new things

The combination of these two factors created a flourishing of new ideas that were based upon Roman and Greek designs but also with something new.

New technologies in terms of masonry and construction---particularly cranes and lifting machines---made it possible to build structures that would not have been feasible for the Romans and Greeks; in addition, many Renaissance architects sought to incorporate new concepts in mathematics into their designs. It was quite common to incorporate spheres, tetrahedrons, and particular mathematical ratios (especially the so-called "golden ratio", 1/2 + sqrt(5)/2). Architects experimented with a variety of new designs and methods of construction, and used recent discoveries in engineering to construct wide open spaces that would not have been able to support themselves without newly invented technologies.

In the end, asking where these ideas came from is like asking where any creative ideas came from: Someone had to think of it first, why not them?