What does the body do with excess protein? If you eat a balanced diet and can get enough protein from these meals, what good are protein shakes?

Metabolism. Excess protein is either metabolized to form glucose(sugar) or fat. By definition a balanced diet should have "enough" protein. If you need to supplement your protein intake with shakes, your aren't looking for a "balanced diet", but rather one slighted towards protein intake - I'm not saying that's good or bad. I would encourage you to focus on natural foods rather than shakes. .
Of little benefit. Excess protein, beyond the body's needs is excreted through the kidneys. If healthy, this is of no major consequence, but in kidney disease, this can accelerate failing of the kidneys. The only PROVEN benefit of high protein intake is with high metabolic exercise, where creatine has been shown to slightly accelerate muscle mass accumulation and help in exercise endurance.