What's the psychological basis for "Look at me when I'm talking to you"?

Communication is not only auditory but gestural and visual, and is illustrated by the McGurk Effect. The combination of auditory and visual information when someone is speaking can lead to the perception of a third sound. So, in a way, when someone says 'look at me when I'm speaking', this is a way to ensure that the listener gets all the information possible for accurate perception, as you may mishear the sound, and without the visual input, perception may become inaccurate. Here's a video on the McGurk Effect

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.

https://blog.joytours.com/2012/12/20/the-role-of-eye-contact-in-different-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

https://en.wikipedia.org/wiki/Affiliative_conflict_theory#Balanced

Persons with learning difficulty, especially if linguistically impeded, would heavily rely on facial expression.

http://do2learn.com/disabilities/CharacteristicsAndStrategies/IntellectualDisability_Characteristics.html

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. :)

"typically developing" people (those who are not in the autism spectrum) needs eye contact itself as a signal of communication as well as they observe gaze, facial expression, and other body language parameters to determine a lot of information.

The lack of ncessity of eye contact may hint the presence of autism spectrum disorder or at least broader autism phenotype.