I never like when the more rational type of intelligence and emotional intelligence are separated from each-other. Or when people talk about excluding emotions from certain types of decisions and information processing. There is a great body of literature with evidence that all these processes are interrelated, also with self-concept and consciousness. I would recommend work from neuroscientist Antonio Damasio who pioneered in this area.
I think what happens is that people who have very good abstract reasoning, logical and problem solving abilities often prefer to apply the same processing to human and social concerns especially when they are young. But it's never devoid of emotions, it's just physiologically impossible unless someone has certain types of brain damage. There is also the phenomenon that when people got criticized for being "too emotional" in early life by parents or others, they develop a behavior that seemly shuts off emotional and human relational concerns. But it's there.
I think developing our emotional intelligence involves improving how we deal with these things in ourselves and especially in social life, and how well it's integrated with other types of information processing, self-expression, and communication.
For me personally, what helps most is learning and improving this from experience, while interacting with the world. I definitely wasn't good at dealing with emotions as a kid and not even interested much, I needed some experiences to recognize how important it is if we want to live a full life. Work in progress of course...