"Apology doesn't mean that you were wrong or the other person was right" suggests that offering an apology doesn't necessarily indicate fault or wrongdoing on either side. It implies that apologies can be made for the sake of the relationship itself.
"It means that your relationship is more valuable than your ego" underscores the idea that apologizing signifies a willingness to prioritize the well-being of the relationship and the emotional connection with the other person over one's own ego or need to be right.
In essence, this quote encourages a mindset of humility and empathy within relationships, where the act of apologizing is seen as a way to maintain and nurture the bond between individuals, even when there is no clear right or wrong. It highlights the importance of valuing the relationship above individual pride.