If you are prepared to accept that $G \to G/K$ is a principal $K$-bundle there is an easy proof. You have that $K$ acts on the homogeneous space $K/H$ so you have an associated fibre bundle
$$
\frac{G \times K/H}{K} \to G/K
$$
with fibre $K/H$. The total space is actually $G/H$. You can construct a fibre bundle isomorphism from this to $G/H$ by
$$
gH \mapsto [g, H]_K
$$
where $[g, H]_K $ is the orbit under the $K$ action $(g, H)k = (gk, k^{-1}H)$. Why is this well defined ? You can check that
$$
ghH \mapsto [gh, H]_K = [g, hH]_K = [g, H]_K
$$
as $H \subset K$. It has an inverse which is
$$
[g, kH]_K \mapsto gkH
$$
This is also well-defined as
$$
[gk_1, k_1^{-1}kH]_K \mapsto gk_1 k_1^{-1} k H = gk H
$$