To transfer the messages database from the Facebook Messenger app to your computer, just go to the Apps section in iExplorer and find the Messenger app in the alphabetical list. Then, double-click on the folder to open it. The messages will be in a file called “orca2.db”, and it’s located in this location:
(Your store numbers may be different)
(NOTE: If you have iOS 8.3, you’ll need to go into the Backups section instead. Please see this article for more information and how to do it: App folders in iOS 8.3 are empty. Why is that?).
Just drag/drop it to your computer and then open the orca2.db file in an SQLite manager app. The contents of the conversations are in the “messages” table in the left panel (4th one down) and in the “text” column of the table.
There are several free SQLite manager apps out there. There’s one in the Mac App Store called SQLite Professional. Or on a PC, the Firefox web browser has a free SQLite Manager add-on which can be opened from the Tools menu once installed: