Most traditions include some form of well-wishing for others, for instance in the form of prayer or visualizations, and both as a wish for their happiness (or salvation, awakening) and rejoicing in their current good fortune.
Exploring it for myself, I find that it has two main effects.
First, a well-wishing for others is also a well-wishing for myself, at a human level. It creates an atmosphere of well-wishing where this human self is included. In a more psychological sense, I also see that others reflect aspects of myself, so the well-wishing for specific others is a well-wishing for those specific aspects of myself mirrored in that person. Through all this, there is a sense of wholeness at a human level, a wide embrace of the many aspects of my human self.
There is less war with myself, less resistance to certain aspects of myself, which allows these aspects to be as they are, and also unfold according to their own process, which – with less resistance – includes healing, maturing and developing.
Then, a well-wishing for others directly influence how I experience and interact with others. I see others as (potentially, inherently) whole, beyond whatever they are expressing. I see that they too, as me, seek happiness and to avoid suffering. I see that they too sometimes act from wounds and confusion. There is a more clarity, recognition, and a more open heart, which in turn influence how I interact with others.