The Cartan structure equations for a connection and various associated 1forms can be checked in a straightforward algebraic manner. But is there a geometric or global significance to the equations can one visualize the proof?
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 ooLie 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 taylormade for your needs. 


Well, this is how I'd think about getting the curvature from a 1form, but this might not be what you were asking about. Suppose we have a connection on a Gbundle. 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 Gbundle 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 1form across the parallelogram. Hope this helps. 

