The "righteousness" of the Beatitudes refers to our personal righteousness. For us to be righteous, to be pure, before God.
Not that I'm worthy to interpret Elder Paisios, who I consider a saint, but to me is seems like he's pointing out that those who say "This person has hurt me and I'll be vindicated by God." actually has pride in himself, and judges the wrongs of the other person, as we are deeming them to be on the receiving end of God's "justice."
While certainly God says "vengeance is mine, I will repay" the point is, such judgments are in fact God's, and it's none of our business to even be thinking about such things, as to do so causes us to fall into pride and judgment, as I said above. Rather, we should focus on ourselves "working out your salvation with fear and trembling." But when it comes to others, Christ tells us to turn the other cheek, and know that we are blessed when we are persecuted. The Fathers teach us that such ridicule is an opportunity for us to accept rebuke and learn true humility, a central and vital virtue. This is especially true of that rebuke or ridicule is unjustly given to us, but our humble acceptance of such rebuke profits us all the more because it is unjust, and we still endure it in Christian love and forgiveness.
I'm not going to pretend these are easy sayings. I myself all too often respond, at least in my heart if not in my thoughts, words and deeds, with anger and pride. This is very hard thing to grasp, harder more to actually practice. But, once we have attained to such a high state and obtained that humility, we will have conquered a host of passions.