Extraversion is among the leadership traits and has two facets:
Extraverted leaders tend to be more successful for several reasons. First, they talk more and are more forceful in their opinion. Second, they take more air time in team discussions and third, they tend to develop a higher number of relationships. Each relationship can serve as a resource when needed.
Talking does tend to be associated with leadership. The person who speaks up is often perceived as the leader. What the leader says is of course, equally important. The comments need to be on point and add to the conversation. The second component, being more forceful in expressing opinions, keeps the discussion focused on one person's agenda. Quantity of talk alone, while a component of extraversion, is not sufficient to bestow leadership qualities.
Developing and maintaining a high number of relationships, increases the scope of influence the leaders has. Knowing a large number of people gives the leader access to thought leaders, influencers and the right person to accomplish a task.
As leadership traits, assertiveness and dominance are "Goldilocks" traits, they must be just right. What is deemed "just right" is different for men and women. Women who behave at the "just right" level for men are often disliked and perceived as too hard. Women leaders have a finer line to walk on this trait than men and must also be willing to be disliked when it's necessary so as to be effective.