The context for this question is the same as that is described in How to see that $M$ is Galois over $k$ in Lang's proof that solvable extensions are a distinguished class? (Prop. VI.7.1, *Algebra*)
However, my doubt is as to why is it that $\sigma K = K$ for all embeddings $\sigma$ of $L$ over $k$? I understand the rest of the proof barring this one step
This is just because $K$ is Galois (in particular, normal) over $k$. So $K$ is a splitting field of some set of polynomials over $k$, and so its image under any embedding into $L$ will be the extension generated by the roots in $L$ of those polynomials.