Let $L/K$ be a finite Galois extension of fields with Galois group $G = Gal(L/K)$. According to the famous Hilbert's 90 we know that the first cohomology vanish:
$$H^1(G, L^*)=\{1\}$$
My question is why holds following generalisation:
$$H^1(G, GL_n(L))=\{1\}$$
My considerations:
We have following exact sequence of algebraic groups
$$0 \to L^* \to GL_n(L) \to PGL_n(L) \to 0$$
Applying $H^*(G,-)$ we conclude that $H^1(G,GL_n(L)) \to H^1(G,PGL_n(L))$is injective.
But why $H^1(G, GL_n(L))=\{1\}$?
Obviously it suffice to show that $H^1(G,PGL_n(L))=\{1\}$ or that the induced map $H^1(G,PGL_n(L)) \to H^2(G, L^*)$ is injective. But why one of these two statements hold?
So, it is true that $H^1(G,\mathrm{GL}_n(L))=0$. One easy way to prove this is to note that this pointed set is classifying vector spaces $V/k$ such that $V_L\cong L^n$. There is only one such vector space.
Note though that $H^1(G,\mathrm{PGL}_n(L))\ne 0$ in general. This is classifying central simple $K$-algebras that become split over $L$, for which there are many for general $K$.
Since the OP knows about algebraic geometry, I can say more. Let $X$ be a scheme and let $\mathrm{GL}_n$ be the normal group scheme associating to an affine $X$-scheme $\mathrm{Spec}(R)$ the group $\mathrm{GL}_n(R)$. Then, the etale cohomology group $H^1_\mathrm{et}(X,\mathrm{GL}_n)$ classifies rank $n$ vector bundles on $X$ up to isomorphism. For a Cech covering $\{U_i\}$ of $X$ in the \'{e}tale topology the Cech cohomology group $\check{H}^1(\{U_i\},\mathrm{GL}_n)$ classifies rank $n$ vector bundles on $X$ which become trivialized on every element of $\{U_i\}$ up to isomorphism. For a discussion of this see page 78 of this.
The relationship now is that $\mathcal{U}=\{\mathrm{Spec}(L)\to\mathrm{Spec}(K)\}$ is a covering of $\mathrm{Spec}(K)$ in the \'{e}tale topology and so $H^1(\mathrm{Gal}(L/K),\mathrm{GL}_n(L))=\check{H}^1(\mathcal{U},\mathrm{GL}_n)$ classifies rank $n$ vector bundles on $\mathrm{Spec}(k)$ which become isomorphic to $L^n$ on $L$. There is only one vector bundle on $\mathrm{Spec}(k)$ up to isomorphism--$k^n$.
The proof of this generalized theorem is just descent theory (in the etale topology) for quasi-coherent sheaves. This is just the (vast) generalization of the Galois descent theory for vector spaces I alluded to in the comments below.