Why do I need to know mentir comme un arracheur de dents?

Because it's a funny old French expression that hasn't lost its validity yet.

What does it mean?

Mentir is French for ‘lie’ and un arracheur de dents is ‘a tooth puller’, so the expression translates to ‘lying like a tooth puller.’

Really, it means you're a big liar.

A tooth puller is old French slang for ‘dentist’, inspired by the old image of dentists as wicked and white-cloaked, armed with huge pincers and eager pull your teeth out. 

 “It won’t hurt,” they would promise. (It always hurt like hell.)

Use it like this

French people most frequently use it about politicians who they believe have not lived up to their promises.

In the tweet below, an MP said about the French President that “Macron has lied like a tooth puller when it comes to the minimum wage.”

You can use it in the upcoming election, if there's a candidate you don't trust:

Jamais je ne voterai pour lui/elle, il/elle ment comme un arracheur de dents. – I'll never vote for him/her, he/she lies like a tooth puller.

But you can use it about anyone who you think is generally untrustworthy.

Ne sors pas avec ce mec, il ment comme un arracheur de dents ! – Don't go out with that guy, he lies like a tooth puller!

Faut arrêter de croire à ses histoires, elle ment comme un arracheur de dents. – You need to stop believing her stories, she lies like a tooth puller.