The overall meaning of this quote is that it's not only a person's body that is important to his or her overall health. Instead, our emotional and spiritual well-being are just as important.
In the quote you give "a doctor can cure the body" refers to the need for physical health. But then the part about a cure for the mind and soul refer to the idea that you need mental/emotional health (cure for the mind) and spiritual health (cure for the soul) as well.
You can see the truth of this statement when you consider instances where people die because they "lose the will to live" on the death of a loved one or some other such tragedy.
William Wordsworth wrote a sonnet relative to this quote entitled, "The World Is Too Much with Us":
The world is too much with us; late and soon,/Getting and spending, we lay waste our powers....
For many, especially those of a poetic soul, the world becomes oppressive. Feeling this oppression of the soul, Lord Byron wrote that melancholy sat over him "like a cloud" that would not break and descend in rain. There must be some solace for a person in such times of melancholy. Often a trip to the mountains or to the beach or other scenic natural spot is very therapeutic to people. Rest is also helpful in healing the troubled soul as is the good company of family and friends. In addition, recreational activities help soothe people's minds and souls. Winston Churchill declared that "what is good for the inside of a man is on the outside of a horse."
Reading poetry, watching a good movie, camping--there are many "cures" for troubled minds and souls. Each person finds his/her own. If none of these work, then the person must seek professional help, for the illnesses of the spirit are as real as the illnesses of the body, and can be just as devasting.