Worrying about how to be “the bigger person” might actually defeat the purpose. There's a huge community of individuals who preach about self-love and why you need to love yourself before you can love others.
This isn't always wrong, however, I think our definition of “love” is off. There's different forms of love, but the word itself has become an enigma to society. What is the one thing that society is best at doing? That's right, twisting words and causing confusion. We're told many lies throughout our lives, and so of course, the concept of love has to be twisted as well.
But, since love is also very powerful, It's mysterious - not completely changed. Lies can only be taken so far. Sometimes, they’re not enough to fool everyone. That's evidence of the first quality of love: power. Yes, love is powerful, but it's only powerful when it's genuine.
No one is perfect, but everyone is trying to be. When in a situation of conflict with others, it's very common for us to say things like “Well, I'm just going to be the bigger person and do XYZ.” It's all backwards. In order for anyone to be “the bigger person” they first have to be genuine - that takes humility.
I'm not great at this. It's very difficult to be genuine when you're hurt, but, the good thing is that love is also a choice. I think the first step is just climbing the obstacles that are keeping you from loving freely.