Martin, http://www.math.uconn.edu/~kconrad/blurbs/galoistheory/galoisdescent.pdf is a handout on this kind of stuff and Theorem 2.14 there gives a proof of the bijection between $K$-forms of $V$ and $G$-structures on $V$.

As for an explicit inverse map, see the top of page 6. Let $Tr_G \colon V \rightarrow V^G$ by $Tr_G(v) = \sum_{\sigma \in G} \sigma(v)$. If $d = [L:K]$ and $\alpha_1,\dots,\alpha_d$ is a $K$-basis of $L$, there exist $\beta_1,\dots,\beta_d$ in $L$ such that $$ v = \sum_{j=1}^d \alpha_jTr_G(\beta_j v) $$ for all $v$ in $V$.

Martin, http://www.math.uconn.edu/~kconrad/blurbs/galoistheory/galoisdescent.pdf is a handout on this kind of stuff and Theorem 2.14 there gives a proof of the bijection between $K$-forms of $V$ and $G$-structures on $V$. It doesn't qualify as "short", and whether it's "nice" or not is too subjective. I wrote it for a target audience that knows only Galois theory and tensor products.

As for an explicit inverse map, see the top of page 6. Let $Tr_G \colon V \rightarrow V^G$ by $Tr_G(v) = \sum_{\sigma \in G} \sigma(v)$. If $d = [L:K]$ and $\alpha_1,\dots,\alpha_d$ is a $K$-basis of $L$, there exist $\beta_1,\dots,\beta_d$ in $L$ such that $$ v = \sum_{j=1}^d \alpha_jTr_G(\beta_j v) $$ for all $v$ in $V$. The right side provides a decomposition coming from $L \otimes_K V^G$.