I'd look in books on representation theory rather than books on homogeneous vector bundles for this sort of thing. Perhaps George Mackey's famous book "Unitary group representations in physics, probability, and number theory" would be of interest, and cover such facts.

To answer your question, it's probably better to consider the finite group case, before the more general Lie group case. So let $H$ be a subgroup of a finite group $G$. Let $(\tau, W)$ be a finite-dimensional irreducible representation of $H$ (it's helpful to give the vector space its own name). Let $\psi$ be a section of the homogeneous bundle over $G/H$ associated to $\tau$. In this finite context, this simply means that $\psi: G \rightarrow W$ is a function which satisfies the identity: $$\psi(gh) = [ \tau(h)](\psi(g)).$$

In other words, $\psi$ is simply a vector in the induced representation $Ind_H^G W$. One may decompose this induced representation into irreducible representations of $G$: $$Ind_H^G W = \bigoplus_{\lambda \in \hat G} V_\lambda^{m_\lambda},$$ where $(\lambda, V_\lambda)$ runs over the irreducible representations of $G$, and $$m_\lambda = dim(Hom_G(V_\lambda, Ind_H^G W)) = dim(Hom_H(V_\lambda, W))$$ is the multiplicity of $V_\lambda$ in the induced representation, or equivalently (by Frobenius reciprocity) $m_\lambda$ is the multiplicity of $W$ in the restriction of $V_\lambda$.

Given such a decomposition, we may decompose the vector $\psi \in Ind_H^G W$ as a tuple (or finite sequence of tuples): $$\psi = (v_\lambda^{(1)}, \ldots, v_\lambda^{(m_\lambda)})_{\lambda \in \hat G}.$$

This is the same, in spirit, to the formula you mentioned. The main difference is that I am averse to choosing bases (unlike many physicists). In fact, I think my dislike of choosing bases (or slight laziness) prevents me from deriving your formula on the nose. Sorry!

But to finish my answer, I should say that the entire argument above can be carried out with few changes, when working with a compact Lie group $G$ and closed subgroup $H$, and unitary representations throughout. (This situation, I would bet, is covered by Mackey). A classic example would be when $G = SO(n)$ and $H = SO(n-1)$, so that $G/H$ is homeomorphic to the $(n-1)$-dimensional sphere $S^{n-1}$.

There, when one works with finite-dimensional unitary representations, the Lie group action yields a natural action of the Lie algebra, and hence the universal enveloping algebra. In other words, the representation spaces also have actions of differential operators arising from the groups.

In particular, if one takes an irreducible subrepresentation of $Ind_H^G W$ as before, one gets a space of functions from $G$ to $W$, which is an irreducible representation of $G$ by translation. It follows that the center of the universal enveloping algebra of $G$ acts via scalars on this space of functions. Hence, if $G$ is a simple compact connected Lie group, the Casimir operator (generalization of the Laplacian) acts via a scalar on this space of functions -- i.e., the functions satisfy a nice second order differential equation. This is the source of "spherical harmonics", for example, and the reason for the word harmonic in this context.

I'd look in books on representation theory rather than books on homogeneous vector bundles for this sort of thing. Perhaps George Mackey's famous book "Unitary group representations in physics, probability, and number theory" would be of interest, and cover such facts.

To answer your question, it's probably better to consider the finite group case, before the more general Lie group case. So let $H$ be a subgroup of a finite group $G$. Let $(\tau, W)$ be a finite-dimensional irreducible representation of $H$ (it's helpful to give the vector space its own name). Let $\psi$ be a section of the homogeneous bundle over $G/H$ associated to $\tau$. In this finite context, this simply means that $\psi: G \rightarrow W$ is a function which satisfies the identity: $$\psi(gh) = [ \tau(h^{-1})] (\psi(g)).$$

In other words, $\psi$ is simply a vector in the induced representation $Ind_H^G W$. One may decompose this induced representation into irreducible representations of $G$: $$Ind_H^G W = \bigoplus_{\lambda \in \hat G} V_\lambda^{m_\lambda},$$ where $(\lambda, V_\lambda)$ runs over the irreducible representations of $G$, and $$m_\lambda = dim(Hom_G(V_\lambda, Ind_H^G W)) = dim(Hom_H(V_\lambda, W))$$ is the multiplicity of $V_\lambda$ in the induced representation, or equivalently (by Frobenius reciprocity) $m_\lambda$ is the multiplicity of $W$ in the restriction of $V_\lambda$.

Given such a decomposition, we may decompose the vector $\psi \in Ind_H^G W$ as a tuple (or finite sequence of tuples): $$\psi = (v_\lambda^{(1)}, \ldots, v_\lambda^{(m_\lambda)})_{\lambda \in \hat G}.$$

This is the same, in spirit, to the formula you mentioned. The main difference is that I am averse to choosing bases (unlike many physicists). In fact, I think my dislike of choosing bases (or slight laziness) prevents me from deriving your formula on the nose. Sorry!

But to finish my answer, I should say that the entire argument above can be carried out with few changes, when working with a compact Lie group $G$ and closed subgroup $H$, and unitary representations throughout. (This situation, I would bet, is covered by Mackey). A classic example would be when $G = SO(n)$ and $H = SO(n-1)$, so that $G/H$ is homeomorphic to the $(n-1)$-dimensional sphere $S^{n-1}$.

There, when one works with finite-dimensional unitary representations, the Lie group action yields a natural action of the Lie algebra, and hence the universal enveloping algebra. In other words, the representation spaces also have actions of differential operators arising from the groups.

In particular, if one takes an irreducible subrepresentation of $Ind_H^G W$ as before, one gets a space of functions from $G$ to $W$, which is an irreducible representation of $G$ by translation. It follows that the center of the universal enveloping algebra of $G$ acts via scalars on this space of functions. Hence, if $G$ is a simple compact connected Lie group, the Casimir operator (generalization of the Laplacian) acts via a scalar on this space of functions -- i.e., the functions satisfy a nice second order differential equation. This is the source of "spherical harmonics", for example, and the reason for the word harmonic in this context.