The wisdom behind visiting the sick is to remember the Hereafter, and to warn against imitating the disbelievers in their purpose for visiting the sick.

The Shaykh : However, it is necessary to remind you of that truth I mentioned earlier: that the Prophet (peace and blessings be upon him), when he commanded visiting the sick in some hadiths, explained the reason for it by saying that it . So I want to remind you not to forget your religion, and not to be stamped with the nature of your enemies in creed and religion. For visiting the sick is, in reality, like a natural cosmic law; because every person gets sick, and every person feels that one day he will get sick just as his neighbor does, and he hopes to be visited and cared for as he did with his neighbor and others. But the difference must be between the Muslim who visits the sick and the non-Muslim in achieving that goal which the Lawgiver aimed at in legislating the visitation of the sick, which is remembering the Hereafter.
You know that some religions have adopted the profession of visiting the sick and helping them as a means of proselytizing their religion. This is because their religion has no pillars and no foundations to which they can call people by logic and sound reason. So they exploit serving people who are far from their religion, and among that is helping them in their sickness, and they take that as a means to bring them into their religion. We have recited to you more than once, and you must know this truth to be distinguished by your Islam in action and creed from others. If we imagine that any non-Muslim believes in Allah (Mighty and Majestic), or even more than that, that he worships Allah (Mighty and Majestic), we must remember that he does not believe in Allah (Mighty and Majestic), nor does he worship Allah (Mighty and Majestic), but rather he believes in other than Him and worships other than Him. We have recited to you more than once that noble verse which makes this truth explicit, saying: .
Therefore, the prevailing view today among people, and especially among some Islamic writers, is that this or that type of disbeliever who does not believe are our brothers in religion, our brothers in faith in Allah (Mighty and Majestic), they claim. And they are ignorant, and I do not think they are feigning ignorance, of that truth which the previous verse thundered: .
Therefore, from this remaining reminder, there must be a difference in your starting point, in your performance of what Allah has prescribed for you as obligatory or recommended. Among that is what we are currently discussing, visiting the sick, that our visiting of our sick not be from this door, the door of social hypocrisy. Likewise, it should not be that our following of our funerals is from this same door, but rather it should be for the sake of Allah (Mighty and Majestic) and for the goal which the Wise Lawgiver aimed at. We follow funerals and visit the sick because in that is the pleasure of Allah (Blessed and Exalted is He), and in that is a reminder to us of our end and the outcome of our affair, which is death. We do not say we visit the sick because it is a beautiful and good thing, and this is a customary matter among people. For in the view of Islam, not all people are "people"; rather, the people are those who believed in Allah and His Messenger.
So it is recommended that when we visit the sick, we remember why we visit. As for the assistance that comes, that is a matter of obtaining the already obtained. But the goal which the Muslim should achieve to be distinguished from others is to remember the Hereafter through his visitation of the sick. Here the Muslim is distinguished from the non-Muslim. Otherwise, we know that the Christians visit each other, and even those who have no religion visit their like and others as well. All of that is from the door of this social hypocrisy, and it does not benefit them anything with Allah (Mighty and Majestic). So we should be distinguished in our actions by our legal objectives, and if an action is not accompanied by a legal intention, it has no value in Islam. This is what I want to remind you of on the occasion that it is the right of a Muslim upon a Muslim that if a Muslim gets sick, he should visit him.