Having the emotional capacity to focus on something besides one's own needs. In a social interaction, emotional bandwidth is the ability to divert attention away from how one is being perceived (and one's interests) towards attending to the other person. It means feeling secure enough with who you are that you can focus on another person's verbal and non-verbal language. In a conversation, it's crucial to have enough emotional bandwidth to observe the other person objectively.
An example: Although Victor is a perceptive person, he doesn't have the emotional bandwidth to notice whether people are interested in what he's talking about. If he felt more secure as a person, he would be freed up to listen empathetically to others.