Let me just disagree here: a manager is NOT a psychologist, and shouldn't try to be one. Good understanding of psychology has NOTHING to do with managing people. People skills yes, but that is something entirely different from psychology.
A great psychologist can suck at managing people.
They are entirely different skills, with very little overlap.
I'm going to respectfully disagree with you here and argue that understanding some facets of psychology (perhaps wrapped in a broader definition of people skills) are essential to long term success managing teams and projects.
Specifically, understanding Goal-Setting Theory, Motivation Theory, and Self-Regulation Theory can provide substantial insights into why people think the way they do and give managers a framework to understand their employees and how their actions will impact their motivation and goals. The book PeopleWare touches on this a little bit when they discuss the role of autonomy in improving worker performance. Knowing that people like to feel ownership of their work and projects helps me, as a manager, guide the team and work on framing things in terms of my direct reports exercising autonomy rather than me dictating. They'll be happier and feel more involved in the work and as a result, the quality of work will be higher.
This isn't to say you need to be a life coach, therapist, and parental figure wrapped up in one. That's not your job -- though team member's lives will impact their work at some point and you'll need to empathetic. Your job to is to guide your team to meeting their objectives and understanding part of the human psyche and what motivates people will make you better at this job. Hands down.
I'd add to that some understanding about major personality types/dimensions. Being able to recognize them helps you in negotiations, assigning people to roles etc.
I would also add that it can differentiate between a manager and a leader. Managers can just handle process, but a leader needs to have deeper understanding of individuals and teams. I've met plenty of managers who definitely weren't leaders.
A great psychologist can suck at managing people.
They are entirely different skills, with very little overlap.