I definitely can't explain any better than the quote I have already put in here about how fasting can do in a matter of days that which could and often does take meds several weeks to do. He stated there that this is because fasting effects all the neurotransmitters at once whereas a drug cannot be anywhere near as efficient.
In general though a fast will first of all rebalance the entire body. It will give it a period of perfect nutrient digestion from it's own stores, so if it is a deficiency causing the Depression a fast would help immensely. It also cleanses the entire body, making nutrient absorption post fast much more efficient. A fasting body will not only cleanse but also repair damaged tissues during, the end result being a cleaner, healthier and more energetic body. The brain is part of the body and will receive 'high priority' treatment in all fasts being one of the most important vital organs, and will of course also benefit from the improved condition of the body.