Active Listening.

All of the answers so far represent good advice, but they're primarily minor behavior adjustments. A useful set of do's and don't's. But they're not very "skill-like". You wanted a skill. Active listening is such a skill.

SKILL: "An ability and capacity acquired through deliberate, systematic, and sustained effort to smoothly and adaptively carry out complex activities."

Isaac Cohen came the closest with his answer on "really listening" and "respond in a way that demonstrates (that you heard them)". What he said captures the essence of Active Listening (AL). There are, however, some additional AL features that you need to round out this particular skill. The following overview includes Isaac's advice and the other items.

(From USU Academic Resource Center - paraphrased and condensed)
HEAR

• Pay attention. Look at them. Stop what you're doing. No interruptions.
• Identify and focus on the important information.
  
  INTERPRET
• Paraphrase what you've heard and ask clarifying questions
  
• Look for body language to either confirm or contradict what you understand and ask if something's out of sync. Example: "I just failed my mid-term, but it's no big deal." (strained smile, eyes moist, voice shaky).
  
  EVALUATE
  
• Don't judge or jump to conclusions. Make sure you have all the pertinent info before forming or expressing an opinion.
  
  RESPOND
• Give feedback to let the speaker know that you heard them.

As to the issue of "quickly", the basics of AL you can pick up pretty quick, but, as with any true skill, mastery comes with practice over time.