In k-fold-cross-validation, why do we compute the mean of the metric of each fold
It's ok to compute the global performance on the concatenation of the predictions for all the K folds after running the cross-validation process, it depends on the goal and on the metric (for instance the mean accuracy over the folds gives the same result as the global accuracy, but that's not true for every evaluation measure).

But very often the goal involves not only measuring performance accurately but also measuring the variance of the performance across the folds, in order to detect instability. This can't be done from the concatenation of the predictions, so it's often more convenient to keep the folds results separated.

(this is my interpretation, there might be other reasons)