IIRC (though I could be wrong), Alan's action against "the club and the board (as a whole)" was due to the stadium ban and totally separate from any proceedings the Kibble members of the board had taken (or implied they were taking) regarding Alan's comments. It seems unlikely that Alan dropping the threat of legal action resulted in the decision to lift the ban (at least directly) as that was not the reason for the ban in the first place. More likely that the board has said to Alan "we'll let you back in but we want you to drop the legal action" and Alan has replied "OK, the legal action was only because of the ban". So, in respect of this matter only, Alan gets what he wants without proceeding with legal action. In essence, the board have realised that they haven't got a (legal) leg to stand on regarding the ban and Alan's threat of legal action has made them shit themselves.
The statement's wording is just a bit of spin to save face.