A concrete explanation: If you have a trivial connection, then a constant frame has covariant derivative zero. If you have a non-constant frame, then it can be written as a $GL(n)$-valued function multiplied by the constant section. Then the covariant derivative of the non-constant frame relative to itself can be obtained by differentiating this using the product rule. The Maurer-Cartan forms appear when you do this.

Of course, after you do this, you want to translate this into a much more abstract and sophisticated proof.

A concrete explanation: If you have a trivial connection, then a constant frame has covariant derivative zero. If you have a non-constant frame, then it can be written as a $GL(n)$-valued function multiplied by the constant section. Then the covariant derivative of the non-constant frame relative to itself can be obtained by differentiating this using the product rule. The Maurer-Cartan forms appear when you do this.

Of course, after you do this, you want to translate this into a much more abstract and sophisticated proof.

(But I do consider this to be a reasonable exercise for an advanced graduate student in geometric analysis and therefore at best borderline for MathOverflow. I'm surprised that no one on math.stackexchange.com has helped.)