I think that what the person was asking the question needs is simply a formula that is more explanatory, or a full-blown explanation of the formula. I'm posting this for the sake of others looking for an answer.
Here is how I understand it.
Start with a less abstract loss function, say the MSE.
Once you divide your data set into K subsets, you calculate the MSE where the test set is one of the subsets k and the function f^(-k)(x_i) is calculated over the training set made of all the points minus subset k. You get
MSE(k)=K/N*sum_{all points in subset k} (y_j - f^(-k)(x_i))^2.
Note that to obtain the average you divide by N/K which is the number of points in subset k.
3. Now you take the average over all K subsets, and you obtain:
MSE = 1/K * sum_k MSE(k)
K and K simplify and MSE becomes simply
MSE = 1/N * sum_{all points!} (y_j - f^(-k)(x_i))^2
Note that each point is counted exactly once. That's why instead of f^(-k) you can write like Hastie et al. "f^(-k(i))".
THe extension to a generic loss function should be trivial.
Hope this clarifies.