Emotional Intelligence serves anybody to be more happy and a valuable member of communities. Project managers at large deal with stakeholders, sponsor, client, the project team, users, outside interested parties. They not only are a member of the project community, they are the leader (hopefully), they are watched and evaluated and often become a role model, if they like that or not. That is why integrity and consistency are important, people do not like surprises, and if they shall trust you, you have to give them the security of being predictable and fair.
Project managers for complex projects have to navigate a more diverse stakeholder constituency, with more differing interests and targets, maybe members coming in and going out more frequently (I was on a project which took me 1 day per week to acquire new members). In complex projects the diverse interests and influences of stakeholders require the project manager to exert influence and to borrow power from stakeholders.
Influencing others is the final value of Emotional Intelligence. It requires that you know yourself and can control your emotions, which results in more rational behavior and self-confidence.
Self-awareness helps you close any blind spots, to close the gap between your view of yourself and how others perceive you. It mitigates hubris and may lead you to the most important leadership competency: humility. After you worked on yourself by working in your self-awareness and self-control, you can focus on working with others. As with self-awareness, being aware of others helps you understand reality better, and how the feel and what they perceive.
The word is empathy, key skills are active listening, observing facial expressions and body language and asking questions. And controlling your feeling to jump in with information or judgement (self-control and humility). Empathy cannot be faked, you have to change your mindset to really care about others - in that moment you engage. Women are often more empathic than men, both can and should improve themselves.
The last emotional intelligence competency I call influencing. Like with self-control you now want to control the feelings and decisions by others. Wait a moment - isn't this manipulation and a violation of their freedom? It may be and in reality it is quite often. Think for example marketing ads, political statements, car sales, electioneering. But sometimes influence is positive for the recipients: in growing your kids, training your students, forming a project team or convincing users of the benefits of a new system (called change management). So what is the difference? Ethics. Influencing that is done for an ethical reason usually results in results that are perceived by all as positive. Influencing that violates ethical values are seen as ... unethical and detrimental. So what are the ethical values that are good reasons and should not be violated?
Rushworth-Kidder researched in many countries and found a list that is practically seen in every culture he observed as common human values. The top eight he found, and not given in a particular sequence, are honesty, responsibility, respect, humility, fairness, community, freedom, compassion. Think about it: if these values are preserved you do not object to being influenced.