Madeleine,
Based on the way ENMeval is structured, it does not return predictions from folds. This is because the folds are used solely to derive averages of model performance, and these are used to select optimal model settings to make a final model with using the full dataset. In other words, the evaluation stats are used to determine how optimal the settings are for a full model. If your spatial CV stats are good, it means that those settings (fc, rm) result in models with good transferability, so it makes sense to use them to make a final model with the full dataset and use this as your range prediction.
However, an alternate method is to average the predictions of folds and use this as the range prediction. I have seen some studies do this. I think this may be a valid strategy when doing multiple repetitions of random CV, as the folds will not differ too much and the whole process is kind of like bootstrapping. However, for spatial CV, the folds can be very different, and thus predicting some of them will present very difficult transfer exercises, resulting in poor stats for some withheld folds and better stats for others. Averaging these (some very predictions, some better ones) doesn't make much sense to me. Happy to keep discussing this though.
In sum, ENMeval does not return these to you, but you can simply make them on your own by selecting all the data in the training folds (occurrences and background) and running ENMeval with just these.
Jamie