Why are some Facebook and Craigslist Apartment Listings so cheap?

First, let me dispel a myth: 

Generally speaking, a landlord doesn't pay to rent their apartment through a broker.  The landlord gives the broker permission to promote the listing, and in turn, the broker collects a fee from the tenant.  There are some exceptions.   For example in the case of OP listings, the owner might pay a month's rent to the broker, but that's more common with big buildings where the landlord wants to motivate many agents to see his or her apartments so the building can fill up quickly and stay filled.    In the end, no:  the landlord is not saving a ton of money by staying off the traditional listing services.

Then why is it possible to find a gigantic 1 bedroom apartment for barely $1400 listed on Craigslist or FB marketplace?     

Here are some possible reasons (aside from the owner being a really, really, really nice guy...) 

• Actually, it's a room, not an apartment.   (This is common!)
• It is an illegal basement rental.   That means it can't be rented on market.
• It is an illegal conversion, meaning the owner may have partitioned an apartment and didn't care to get work permit or Certificate of Occupancy.
• It's an illegal sublet.   A great deal, just don't get caught!
• It's located places like Coney Island, Canarsie or Far Rockaway (great places, but pretty far out!)
• It's a scam in the form of a hijacked listing. 
• Bait and switch via a blind ad.

Finally, you may have an owner who isn't a big fan of fair housing law.  The person may have previously used an agent, but was asking the agent to do things that could result in fines or suspension of his or her license.  As a result, the agent backed out of the agreement, and the owner decided to rent the place out on his or her own. 

Things to look for in ads to vet scams from the real deal:

• Disclosure:  If listed by an agent, the agent's license status and brokerage are provided.  
• Transparency:  You can confirm who the building owner is and contact him or her directly.
• Market Value:   hopefully a good deal, but not a too good to be true deal.
• Photos:  Be wary of no windows, or windows too small for a person to climb out of in the event of flooding.
• Photos suggesting unorthodox construction work (hanging wires, strange plumbing schemes, temporary walls, etc.)

(This list is partial.)

Additional Resources:

Examples of Illegal Rentals
source:  NYcourts.gov

Note:  I am not a lawyer and do not provide legal advice.