I dunno about the explicit inverse, but there are two simple ways I know of showing the map is an isomorpism. The first is just to apply Grothendieck's faithfully flat descent theory to L/K -- one identifies the descent data on an L-vector space as exactly the kind of Galois action you describe. The other, maybe more down-to-earth, is by considering twisted group K-algebra L{G} built so that modules over it are exactly L-vector spaces with the kind of G-action you describe. Here is the key:

Claim: Every finite-dimensional L{G}-module is a direct sum of copies of the module L with its Galois action.

Proof: The natural map L{G} --> End_{K-vect}(L) is injective by linear independence of automorphsims hence an isomorphism by dimension count, and the corresponding fact for the matrix algebra End_{K-vect}(L) is well-known (e.g. by Morita).

Given this the reason that your map is an isomorphism is that by the claim we can reduce to V=L where it's just Galois theory (a direct limit argument reduces to the finite-dimensional case, or we could remove "finite-dimensional" from the above claim using Choice).

Algèbre et théories galoisiennesby Douady and Douady. – Pierre-Yves Gaillard Jun 14 '10 at 5:43