For this achievement you'll need to be using a showman class wrestler, they're the only wrestlers who have the ability to steal an opponents taunt.
Once you're setup in a match and ready to go for this achievement, either get your opponent on the ground or get out of the ring and make some space between you guys because if they interrupt you it doesn't count that taunt toward the achievement. When you want to steal a taunt hold down "Y" and then use the directional buttons on the D-pad to steal a taunt, once you've stolen 3 taunts the achievement will unlock.
An example of this can be seen in the video below.
* Credit to xGamerNetworkx for the video