Water the burned-out area thoroughly as soon as you notice it. It may be that the damage has simply been caused by lack of water over a prolonged period of time and can be restored through hydration. The water will also help wash away excess nitrogen if that is what caused the burn.
Dig up the burnt spot with a shovel or gardening spade. Make sure you remove all the grass as well as 2 to 4 inches of soil.
Replace the soil you removed with fresh soil -- either purchased soil or soil from elsewhere on your property -- and mix in compost or similar materials if you have them available. This will help set up the new soil for grass growth.
Seed the new soil with your grass seed according to the directions on the seed package. Rake the seeds into the soil and mix them with a small amount of starter fertilizer for best results.
Water the area and wait for the grass to grow back in.