First let me describe the norm in degree 0. Let $k$ be a commutative ring, given $\alpha\in H^0(BH,k)$, which corresponds to a an $H$ equivariant homotopy class of a map $EH_+\rightarrow k$ where $k$ has the trivial $H$ action. W can take the $n$th smash power of this map to obtain \begin{equation*} (EH_+)^{\wedge n}\cong E(H^n)_+\rightarrow k^{\wedge n}. \end{equation*} This map is naturally $\Sigma_n \wr H$ equivariant. Using the (equivariant) multiplication in $k$ we can make this map land in $k$ again. Now use the monomial representation above $G\rightarrow S_{G/H}\wr H\cong \Sigma_n \wr H$, we can regard this $\Sigma_n \wr H$ equivariant map as a $G$ map and $G$ acts freely on the source. Taking orbits we obtain an ordinary map $[BG_+, k]$ as desired.
To move up to higher degrees, I find it helpful to use equivariant spectra. So now our map $\alpha$ in degree $i$ will corresponde to a map $\Sigma^\infty_+ EH \rightarrow S^i \wedge Hk$. We proceed exactly as above but with one little snag. After taking smash powers the $S^i$ becomes $S^{i\rho}$ where $\rho$ is the standard $n$ dimensional representation of $\Sigma_n$ ($H$ acted trivially on $S^i$). When $i$ is even and $k$ is a field of characteristic $p$ (all $i$ in characteristic $2$) $S^{i\rho} \wedge Hk\simeq S^{in}\wedge Hk$ equivariantly which lands us in the correct degree. More generally we obtain $RO(G)$ graded coefficients.
For more details on the norm in spectra consult the article by May and Greenlees or Schwede's course notes especially chapter 7.