Do you have to explain what your emergency is?
The order is "Aviate, Navigate, Communicate".

As a pilot, your first responsibility is to keep the plane flying. After that it is to avoid hitting something or to avoid getting lost. After that comes communication. There are many instances of in-flight emergencies where the pilots never talked to anybody about it, because they were busy flying the plane...


  According to FAA, is it required to detail/explain (maybe file some kind of official document) your emergency to some kind of authority?


The FAA has many procedures on reporting accidents, incidents, and emergencies. Whether or not you report it depends on what the emergency was. If its a very sick passenger, there may be no reporting required. If your control cables failed, you are required to report it. I'll update this answer with the regulations as soon as I find them all. Quick answer is any time you declare an emergency or "PAN-PAN", call the FSDO and let them know. Also depending on the nature of the accident/incident, you may be required to immediately notify the NTSB. See CFR 49 830.5


  If you do have to file an explanation, what information do you have to supply?


This depends again on the nature of the emergency, and the FSDO/NTSB will ask you for the proper information when you file the report. At the very least they are probably going to want a narrative of what happened, and depending on what took place, you may want to contact your lawyer first. (Note: You may have to contact the NTSB immediately but it doesn't say you need to provide details, just that there was an incident, so the NTSB would be your first call "I had an incident", short and sweet, and the lawyer second.)


  Also what would happen if they "didn't believe" that situation was an emergency?


Its not ATC's responsibility to determine if you are lying or not. When you as a pilot say "emergency" you have their full attention. After you get on the ground and the emergency turns out to be you forgot to call your girlfriend/wife and wanted to get ahead of other traffic, then the FAA will get involved. Until you are on the ground, an emergency is handled as priority, regardless of the reason.



Here you can read a good story about a stupid pilot who abused the "emergency" system by landing on a public beach in New York because of an airsick passenger and an engine that was "running a teensy bit rough". He didn't officially declare an emergency, and I can't find any record of official enforcement action against him for doing that. Note the NTSB did not investigate this "incident".