Why do I need to know tomber dans le panneau?

Because if a French person says you've done this, then it's not good.

What does it mean?

It means to fall into a trap, so it's a synonym for the more literal expression tomber dans le piège or se faire piéger (to be trapped). More simply, it can be used to mean fall for something such as a scam, a trick or a practical joke (such as the annual poisson d'avril). 

What's the story behind it?

While your French dictionary will tell you that panneau means sign or panel, its use here refers to something else.

Back in the olden days, probably as far back as medieval times, a panneau was net or fabric which was stretched by hunters over the trails made by wild animals like rabbits in order to catch them, so it's easy to see where the phrase comes from. 

Use it like this

Je lui ai expliqué que je n'ai aucune intention de tomber dans le panneau –   I explained to him I have no intention of falling into the trap

On ne voudrait pas que d’autres gens tombent dans le panneau – We wouldn't want other people to fall for it