A Facebook app (Facebook, Facebook Messenger, etc.) has to use the internet, so you need either a wifi connection or a mobile data connection. Facebook itself is free. A mobile data connection isn't. A wifi connection may be, depending on where you are, what kind of service you have, what business near you provide free wifi, etc.
Ther's a difference between paying a service to use it (to read the Washington Post online, you have to pay the Washington post - to read this forum [or post to it] you don't have to pay - Android Central is free).
It's like an amusement park - the park may be free, even the rides may be free, but getting there will cost you money - the money has nothing to do with the park. You pay for internet service, whether wifi or data, but Facebook is free.