What is the geometric significance of Cartan's structure equations? The Cartan structure equations for a connection and various associated 1-forms can be checked in a straightforward algebraic manner.  But is there a geometric or global significance to the equations- can one visualize the proof?
 A: Well, this is how I'd think about getting the curvature from a 1-form, but this might not be
what you were asking about.  
Suppose we have a connection on a G-bundle. Curvature
measures what happens when we transport around a small loop. We can take this to be a parallelogram: move in direction X, direction Y, then direction -X, then direction -Y.  
Now, once we have fixed a trivialization of the G-bundle at every points, the effect of moving in direction X is described by an element gX in G, close to the identity.
 The effect of the parallelogram motion is g_X g_Y g_X^{-1} g_Y^{-1}, but corrected slightly (because
the effect of moving along edge -X is not the inverse of moving along X; the two edges are displaced.)
The former term corresponds in usual notation to [\omega, \omega]: it measures the noncommutativity of G. The "correction" corresponds to d\omega: it measures the variation in the 1-form across the parallelogram.  Hope this helps. 
A: There is a nice grand story behind all this. I don't know if you like thinking that way, but things do clarify when one looks at it from a more general perspective of oo-Lie algebroid valued differential forms with curvature.
I am still working on these entries, trying to expose some of my work with Jim Stasheff and Hisham Sati. If you bug me with questions, there is a good chance that I'll improve the exposition taylor-made for your needs.
