dg.differential geometry - What is the origin of the formula for the Lie derivative along a Killing vector?

Background

Let $(M,g)$ be an $n$-dimensioal riemannian manifold. A vector field $X$ on $M$ is said to be a Killing vector if the flow it generates is an isometry; that is, it preserves the metric $g$. There are many ways of writing this. The one which is relevant for this question is the following. If we let $nabla$ denote the Levi-Civita connection, then $X$ is Killing if and only if the endomorphism $A_X : TM to TM$ defined by
$$A_X(Y) = - nabla_Y X$$
is skewsymmetric, so that for all vector fields $Y,Z$ on $M$ one has that
$$ g(A_X(Y),Z) = - g(Y,A_X(Z)).$$

In summary, if we let $mathfrak{so}(TM)$ denote the bundle of skewsymmetric endomorphisms of $TM$, then $X$ is Killing if and only if $A_X$ defines a section of $mathfrak{so}(TM)$.

Let $mathrm{SO}(TM)$ denote the bundle of oriented orthonormal frames of $TM$. It is a principal $mathrm{SO}(n)$ bundle over $M$. In the case I'm mostly interested in, $M$ is a spin manifold, so that there is a principal $mathrm{Spin}(n)$ bundle $mathrm{Spin}(TM)$ and a bundle surjection $mathrm{Spin}(TM) to mathrm{SO}(TM)$ which restricts fibrewise to the covering homomorphism $mathrm{Spin}(n) to mathrm{SO}(n)$.

If $rho : mathrm{Spin}(n) to mathrm{GL}(V)$ is a representation, then we can form the associated vector bundle
$$E := mathrm{Spin}(TM) times_rho V.$$
Attached to every Killing vector $X$ on $M$ we have a Lie derivative $mathcal{L}_X$ on sections of $E$. Explicitly, this Lie derivative takes the form
$$ mathcal{L}_X sigma = nabla_X sigma + rho(A_X) sigma,$$
where I am using $rho : mathfrak{so}(n) to mathfrak{gl}(V)$ also to denote the derivative map of the the representation. (I am also identifying $mathfrak{so}(TM)$ with $mathfrak{so}(n)$ via a choice of local frame.)

For example, in the case of the tangent bundle itself viewed as an associated bundle where $rho$ is the defining representation of $mathfrak{so}(TM)$, then as expected, we find
$$ mathcal{L}_X Y = nabla_X Y + A_X(Y) = nabla_X Y - nabla_Y X = [X,Y] .$$

Question

Although I quite often use the formula for the Lie derivative $mathcal{L}_X$ along a Killing vector, I do not feel I have a good conceptual understanding of it.

Could someone enlighten me?