A bout, literally is a part of a fight. When two people fight in boxing the game is divided into rounds, or bouts. But it is generally used to suggest a short period of intense activity.
When used figuratively, it means a short period of time when the person is undergoing great stress or adversity.
For example, we could say 'I had a bout of the flu', meaning that i had a few days where my health was not good because of an influenza infection.