See @sharpe2000differential page 184.
Definition
Consider a Cartan geometry $(P,A)$ modeled on $(G,H)$, and let $\Omega$ be its curvature. If
$$ \rho: \mathfrak{g} \to \mathfrak{g}/\mathfrak{h} $$is the canonical projection then $\rho(\Omega)$ is called the torsion.
$\blacksquare$
It is a measure of the deviation of the curvature from belonging to $\mathfrak{h}$. The idea, I think, is that if we travel through a closed loop, if the geometry is not flat we don't recover our original "position+orientation", but a modified "position+orientation" due to the curvature. If after our travel we only have modified our "orientation" then we are in a torsion-free Cartan geometry. The change of the "position" (a translation) is the torsion.
Update: think about this: in a reductive Cartan geometry with torsion free Cartan connection then the curvature of the Cartan connection is practically the same as the curvature as the curvature of the principal connection.
(I have to review all this)
In this question on MO it is said that the torsion $\rho(\Omega)$ can be computed as
$$ de+\omega \wedge e $$where $A=\omega+e$, being $\omega$ the projection on $\mathfrak{h}$ (a kind of "rotations") and $e$ the projection on $\mathfrak{g}/\mathfrak{h}$ (a kind of "translations"). Moreover, it is said that $\Omega=d\omega+\omega\wedge \omega$... This is what is called in @needham2021visual page 440 the Cartan's second structural equation.
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Author of the notes: Antonio J. Pan-Collantes
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