A Cliffhanger occurs when both you and your opponent run out of ego points at the same time. Think of it as an overtime round where the person who comes up with the best single roast wins.
I got this while going for using 3 comebacks in a single battle. Since I wasn't firing on all cylinders trying to build my comeback meter and using less than optimal insults, we both ended up dying during the same round, which took us to the Cliffhanger. When in the Cliffhanger, make sure to use every possible phrase you can, utilizing your 'ands' and 'buts' to make the better insult, because whoever gets more points in this round automatically wins the battle. Assuming you win the Cliffhanger, the achievement will pop when the characters are doing their outro dialogue.