What you are asking for is an introduction to the theory of $G$-structures, for which the (Maurer-Cartan) structure equations are a basic tool.

There are many sources for this material, starting, of course, with the fundamental works of Élie Cartan on the subject, though many find his expositors, who use more modern language, easier to follow.

S.-S. Chern has an article, *The geometry of G-structures.*
Bull. Amer. Math. Soc. **72** (1966) 167–219, that I highly recommend as an introduction. There is Sternberg's book, *Lectures on differential geometry*, which is a good reference towards the end of the book. A more modern treatment specifically on the use of structure equations in modern differential geometry is R. Sharpe's book *Differential geometry. Cartan's generalization of Klein's Erlangen program.* Graduate Texts in Mathematics **166**. Springer-Verlag, New York, 1997. There is also the book by Ivey and Landsberg *Cartan for beginners: differential geometry via moving frames and exterior differential systems.*
Graduate Studies in Mathematics **61**. AMS, Providence, RI, 2003.

That's just a small sample, but it should be enough to give you a choice of authors so that you can figure out which comes closest to being able to help you.