This is summarising pages 30 - 33 in The Topology of Fibre Bundles by Steenrod. I will omit proofs as they are quite long. There is a Corollary on p. 31 (to a Theorem on p. 30), which is relevant to your question:
Let $H$ be a closed subgroup of $B$ (not assuming $B$ is a Lie group yet, just a topological group), and consider the projection $p:B \rightarrow B/G$. If there exists a section $s:B/G \rightarrow B$, then $B$ is a fibre bundle over $B/G$ which assigns to each $b \in B$ the coset $bG$. The fibre of the bundle is $G$ and the group is $G$ acting on the fibre by left translations. (Proof is on p. 31 - 32).
Now introducing Lie groups, a requirement following this construction that a section $s:G/H \rightarrow G$ exists is treated in p. 110, Prop. 1, of Theory of Lie Groups by C. Chevalley. We also require that the maps are now smooth and that the projection map $p:G \rightarrow G/H$ is of maximal rank everywhere in $G$ (again discussed in Steenrod). Then the above theorem applies to Lie groups too.