There are formulae for computing the leave-one-out cross-validation error in closed form for many models, including least-squares regression, but as far as I am aware there isn't a general formula for k-fold cross-validation (or at least it may be possible but the computational advantage is too small to be worthwhile).
The formula in the book isn't saying very much it is just saying that the cross-validation error is the average of the loss function (L) evaluated using models trained on different subsets of the data. The superscript $-\kappa(i)$ just means "model $f$ is trained without the training patterns in the same partition of the dataset as pattern $i$". Sometimes writing things in formal mathematical notation makes things less ambiguous, but it doesn't necessarily make it any easier to understand than the text - I think this is one of those occasions.