This achievement is similar to the "Lady Helper" achievement. Stinky Bugs will appear every 4-5 hours. To get rid of a Stinky Bug, you must tap them 3 different times. After each tap, the Stinky Bug will move to another location. After 3 taps, they will fall over and disappear. This will count as getting rid of one Stinky Bug.
Note: There are glitches related to the Stinky Bugs. One glitch can cause the Stinky Bugs to stop appearing in the game. If this happens to you, please check your village very carefully for them and consult with this thread. If you do not find any in your search and they still do not show up, you will eventually need to start the game over. You will also need to start your game over if you run into another Stinky Bug glitch. This glitch happens when you are get rid of your 100th Stinky Bug. Instead of getting the achievement, your game will crash. Please see the Roadmap and this thread for information on what to do if these glitches happen to you.