Use an aeration tool to pull plugs from the yard. This will remove soil and create better drainage in the muddy area. Although this may be the only thing you need to do, if you have a significant amount of clay in the soil this is only the first step in the fix process.
Spread sphagnum or manure over the muddy area and rototill. By mixing in these soil amendments, the drainage will be immediately improved. The sphagnum and manure both contain larger air pockets than clay so they drain water faster.
Dig a French drain system in the muddy area. Dig out a trench about 2 inches wide and the length of the poorly draining area. Fill the bottom 4 inches with pea gravel and cap it with flat stone. Fill the trench back in. This creates an artificial drain in the field.
Plant water-loving pants in the area if there is no way to amend the soil. This can happen in places where the area is close to the groundwater level.