I recently came to know a theorem of Serre, which I couldn't search online with proof.
If $G$ is torsion-free and contains a free subgroup of finite index, then $G$ is free.
Can one provide a reference for proof of this theorem? It would be better if one suggests a book containing possibly simple proofs of this theorem.
Here is a partial answer, covering only the finitely generated case.
Look up the theorem of Magnus, Karass, and Solitar which says that if $G$ is finitely generated and contains a free subgroup of finite index then $G$ is the fundamental group of a finite graph of groups with finite edge and vertex groups. This is also a corollary of Stallings "Ends theorem", but historically the Magnus-Karass-Solitar theorem preceded the ends theorem.
If in addition $G$ is torsion free, it follows that the edge groups and vertex groups are all trivial, and so $G$ is free.