(condensed from Ein Ayah, Shabbat 1:43)

Gemara: One who enters to visit the sick on Shabbat, says: "It is Shabbat, so that crying out cannot be done and healing shall come soon."

Ein Ayah: The existence of diseases plays a positive role in shaping a person’s character. First, it makes him "surrender" to Hashem. Second, the knowledge that people can become sick softens the stubbornness of the stubborn and the wickedness of the wicked.
In order to complete the personal improvement brought on by illness, one needs to engage in prayer to Hashem to save him from his difficult situation. Such prayers are very important for a person and are one of the reasons that Divine Providence arranges disease, by encouraging one to return to Hashem (see Tehillim 90:3). When the sick person or those close to him have prayed intensely and thereby actualized the goal that the illness improves their moral state, and hearts have been brought closer to Hashem, the person may recover, given that the goal was met.
There are two ways in which difficult times can improve a person’s ways. The simple one is that the situation will give him the emotional impetus to turn to Hashem and strengthen himself spiritually and morally.
A higher level is that instead of crying out to Hashem to save him, he purifies his thinking and recognizes how proper it is to rely on Hashem in silence. He should realize that man does not know what is good for him and that even that which seems bad contains hidden good. Illness can actually cause him to focus on the desire that whatever Hashem wants for His world should occur, including his own illness. This is the level of the truly pious, who trust Hashem totally and do not ask for a thing for themselves, because they trust that Hashem will do that which is good for them and for the world, even if it includes their pain. This attitude will lower the desire for powerful prayer regarding their own condition, for why should one cry out when things are good?
The proper approach for a normal person is the first one, in which he maintains his natural desire for life, good health, and happiness. Based on those feelings, he will use his mouth to turn to Hashem in his sorrow and, through that, healing will come. While it represents a lower spiritual level, it is the approach that is closest to him and appropriate for all but the most unique of pious people who care only about what Hashem wants.
The second, select group of people do not need to purify their emotions through prayer at the time of troubles. Rather, troubles can only serve to straighten their understanding of the good in everything that Hashem does, which grows as their troubles grow.
During the week, a person is generally only capable of taking the approach of improving his emotions when turning to Hashem in impassioned prayer. However, on Shabbat the heart and the soul are elevated, and therefore it is a time in which one should not awaken the emotion of crying out to Hashem but should focus on the belief that Hashem has the power and the goodness to do what is best. For that reason, crying out in prayer is forbidden on Shabbat.
A person should not think that without crying out to change the situation, but just through the approach of belief, he will not succeed in becoming healed quickly. The speed of the positive effect of prayer is a function of the degree to which the disease brought the moral goal for which it was designed. If this is so for the lower level response to adversity, then it is all the more so that once one reaches new heights in his level of belief, the healing can come quickly. Therefore, since Shabbat is not the time for crying out, there is every reason to believe that the "healing shall come soon."