Like cockfighting (that I don't recommend to anyone IRL.. Poor roosters), this is the Chocobo Fighting Pit. This place is put together like a mini football stadium/colluseum, bleechers surrounding the pit. Except loads more scummy and only lit up with mako-electric lights around the actual pit area.
Here is where they train chocobos to be aggressive and pit them against eachother, against randomly caught monsters, and against 'traitors' in Midgar Society for entertainment. Sound familiar? If not, do your history.
Tickets are 10 gil each.