Although there are no guarantees, you tend to be happier in a happy environment where people are relatively peaceful and content. On the other hand, it is more difficult for you to be happy while in conflict with your environment, with others.
By widening your scope of interest and looking after everyone, including yourself of course, you both avoid conflict and create better conditions for your own naturally happy state of mind.
This is compassion, what the Dalai Lama calls beneficial egoism.