Sphering ( or whitening ) the data ($X$) means applying a transformation so that in the new basis, the covariance for sphered data ($X^{*}$) is the identity matrix, i.e. $E[X^{*T}X^{*}]=I_{n}$ .
We operate this transformation to obtain significantly simpler computation. As mentioned in 4.3.2 the ingredients of $\delta_{k}(x)$ are
$(x − \hat{\mu_{k}})^{T}\hat{\Sigma}_{k}^{-1}(x − \hat{\mu_{k}}) = [U^{T}_{k} (x − \hat{\mu_{k}})]^{T}D_{k}^{-1}[U^{T}_{k} (x − \hat{\mu_{k}})]$
$\log{|\hat{\Sigma}_{k}|}=\sum_{l}\log{d_{kl}}$
where $\hat{\Sigma}_{k}=U_{k}D_{k}U_{k}^{T}$ is the eigen-decomposition for each $\hat{\Sigma}_{k}$, $U_{k}$ is $p \times p$ orthonomal and $D_{k}$ a diagonal matrix of positive eigenvalues $d_{kl}$.
Let's write the sphering of the data :
$[U^{T}_{k} (x − \hat{\mu_{k}})]^{T}D_{k}^{-1}[U^{T}_{k} (x − \hat{\mu_{k}})]$
$=(U^{T}_{k}x)^{T}D_{k}^{-1/2}D_{k}^{-1/2}U^{T}_{k}x + (U^{T}_{k}\hat{\mu}_{k})^{T}D_{k}^{-1/2}D_{k}^{-1/2}U^{T}_{k}\hat{\mu}_{k}-(U^{T}_{k}\hat{\mu}_{k})^{T}D_{k}^{-1/2}D_{k}^{-1/2}U^{T}_{k}x-(U^{T}_{k}x)^{T}D_{k}^{-1/2}D_{k}^{-1/2}U^{T}_{k}\hat{\mu}_{k}$
We operate the suggested change of variables, $X^{*}\leftarrow D^{-1/2}U^{T}X$ and similarly $\hat{\mu}_{k}^{*}\leftarrow D^{-1/2}U^{T}\hat{\mu}_{k}$. The previous calculation transforms into
$(x^{*}-\hat{\mu}_{k}^{*})^{T}(x^{*}-\hat{\mu}_{k}^{*})=\|x^{*}-\hat{\mu}_{k}^{*}\|^{2}$
Minimizing the discriminant needs to minimize this quantity, that is to say finding the class $k$ minimizing the distance between data and the centroid of class $k$ in the new base.
The last term in $\delta_{k}(x)$ is $\log{\mu_{k}}$ hence the mention on the influence of prior probability $\mu_{k}$.
