As to answering why a person insists on eye contact, the matter is most probably individual. This is the person's preference. Eye contact does not have a universal meaning. Here is a post on differences between Western and Eastern cultures.
Also if to talk "Westerners alone", eye contact can be associated with friendship, sexual attraction, as well as hate and struggle for dominance. See Wikipedia on the Affilliative Conflict Theory
Persons with learning difficulty, especially if linguistically impeded, would heavily rely on facial expression.
Scholars discussing ideas may "take turns" in eye contact, generally keeping one another in visual field. Focus tends to shift attention to hearing.
Well, everyone needs to balance his or her approach individually. :)