In Gauge Theory and Variational Principles by David Bleecker, the curvature of a $\mathfrak{g}$-valued 1-form connection is defined as $\Omega^\omega:=D^\omega\omega=(d\omega)^H$, where $\omega(d\omega)^H=0$, where $D^\omega$ is the exterior covariant derivative. However, I do not see where this comes from. My understanding is that the curvature of a field comes from quantifying the holonomy on a manifold. I assume this does something similar, but I do not understand the full details.
In short, I am asking what the motivation for $\Omega^\omega:=D^\omega\omega$ is. Where does this definition and the definition of $D^\omega\phi=(d\phi)^H$ come from?
The curvature is quantified by the holonomy group, you can understand this curvature quantity as an infinitesimal holonomy. This is is spelt out in detail in the Ambrose-Singer theorem. The wiki page only briefly discusses it and gives the example of an affine connection, however there are plenty of resources available on the theorem.
In short, an infinitesimal holonomy is described as the derivative of the holonomy operator at the identity, as usual when working with Lie group the "infinitesimal generators" are in the Lie algebra. We know the holonomy operators are a sequence of parallel transports in a loop which are locally exponentials of the $\mathfrak{g}$-valued 1-form connection.
By parametrizing the loop $\gamma_s(t)$ in two parameters $(s,t)$ such that one follows the path going from $(0,0)$ to $(1,0)$ to $(1,1)$ to $(0,1)$ and finally back to $(0,0)$, we can know note that the holonomy will be a function $h_\gamma(s,t)$. The result of the computation gives the following:
$$\frac{\partial^2 h_\gamma}{\partial s \partial t}\bigg\lvert_{(s,t)=(0,0)}=Ω^ω(\partial_t \gamma_s(t),\partial_s\gamma_s(t))\bigg\lvert_{(s,t)=(0,0)}$$