Let $M$ be a discrete $G$-module and $M'$ another $G$-module defined by the exact sequence \begin{equation} 0 \to M \to \text{Ind}_G(M)\to M'\to 0\end{equation}
where $\text{Ind}_G(M):=\text{Maps}(G,M)$ is the collection of continuous functions $\alpha:G\to M$ and is called the induced G-module. As stated on page 32 of the book "Cohomology of number fields" by Neukirch-Schmidt-Wingberg, dimension shifting is a technique by which "definitions and proofs concerning the cohomology groups
for all G-modules M and all n, may be reduced to a single dimension n, e.g.
n = 0". Basically, if we take the long exact sequence corresponding to the sequence given above, we get a map
\begin{equation}H^n(G,M')\to H^{n+1}(G,M)\end{equation}
which is a surjection for $n=0$ and an isomorphism for $n>0$ by a theorem in the book (in fact, something more general is true but I think understanding this special case is enough to understand the general case).
My question is how does this help us actually calculate $H^n(G,M)$ for $n>0$. I don't understand how calculating the cohomology of $M'$ is supposed to be easier. It would be great if someone could also give an explicit example where this technique is useful.
The point is that if you know something like $H^1(G, M)$ vanishes for all $M$, then you get this for the higher cohomology as well. In general, for the reason you observed, dimension-shifting is most useful when you have "for all $M$" type statements since you have very little control over $M'$.