At the heart of Emotional Intelligence are three R's: Recognize, Read, Respond.
Recognizing (Self-Awareness) your own tendencies - strengths, weaknesses, moods, drives, emotional states;
Reading (Situational-Awareness) the verbal and non-verbal signals of your environment to accurately assess it and its response to you; and
Responding in ways that are most appropriate based on the evidence gathered around you and within you.
When this is applied to managing others, powerful leaders (high EI) are able to recognize their downstream impact on others, recognizing when their own moods are influencing others' performance and their ability to voice ideas or challenges to them, and recognizing when they must own their disruptive (albeit often unintentional) behavior when they need to and take the necessary steps to adjust, repair and learn from the situation.
He or she is able to read the emotional make-up of their direct reports and connect with them on a level that meets the follower's needs, which may or not be the same style as the leaders'. They do this through empathy, seeing each employee as an individual and complex person with unique talents, needs and perspectives. Which is the opposite of one-size-fits all leadership. He or she takes responsibility for responding in appropriate ways - by not taking out bad moods or misdirected or exaggerated performance intensity on those around him or her. And by providing praise when and how someone needs to hear it and clearly communicating expectations and desired outcomes instead of doing management by mindreading and assumptions.
Leaders with high EQ are not just "yes" people. We have all had to hear bad news, and the way it is delivered is critical. EQ helps you to be fair but firm, assertive and sensitive. It allows you to show care while holding people accountable because you have made enough of an emotional investment in them to drive their engagement and loyalty to you; approaching each employee as a unique asset and resource, taking the time to tap into their internal motivations, passions and talents. A leader with high Emotional Intelligence doesn't do this spontaneously or in a vacuum, instead, they mindfully practice "learning" their employees as a daily habit. Most who have learned to do this report it is their highest leverage leadership behavior - an important shift with huge impact for all. So, I guess it is true, one might say leaders with high EQ do give people exactly what they want.