"Following a review of the manuscript by the editorial board, we have regretfully decided not to consider this work for publication. We thank you for your interest in our journal and..."
This sounds like an editorial ("desk") reject more than anything else. Hence, there typically is no formal, written review that the editor could forward to you. It is just that the handling editor and/or the Editor-in-Chief have decided that the paper is either of low enough quality, or so clearly out of scope, that running it through the full-blown peer-review process would be a waste of reviewer time. While this is of course a harsh judgement for your submission, the editors are entirely allowed to do this - there is no formal "obligation" that any submission will have to be peer-reviewed before it can be rejected.
What can I do in this case? Note that the only thing I want is to know reasons for the rejection.
Realistically, not much. Of course it would be nice if the editors at least gave you some informal pointers why your submission was desk-rejected, e.g., "I am sorry but your submission is out of scope for this journal", but maybe if your request was combative enough, the editor decided that she/he rather did not want to get into an argument with you about this. Anyway, I doubt that there is an obligation on the editor's side to always fully justify each rejection. At the end of the day, acceptance of papers is always a discretionary decision by the Editor-in-Chief, and not something you can formally object to.