"Sickness where guilt is absent cannot come, for it is but another form of guilt" - ACIM from "Only salvation can be said to cure".
Sickness is a form of guilt. You cannot recognize it as guilt if you do not clearly recognize what salvation is, ie the truth of the Atonement. You need to be able to be clear about what guilt IS, and what it is not, ie that it is not innocence, in order to clearly see sickness and recognize it as a form of guilt.
Accepting the atonement for yourself, which is accepting the "cure" for yourself, allows you to recognize what the truth is, on its terms, without your ego distorting it. In your willingness to accept it is true, to understand why it is true, and to experience its truth, it shines a light which reframes the illusion of sickness when seen in others.
Sickness is then more clearly recognized not as a valid state of truth, but as a state of guilt. A state of self attack. A state of punishment. If sickness is present, the person IS attacking themselves, and there is only one reason to do so - they believe that they sinned. Their sickness is literally a symptom of guilt.
Those who believe in sin do not believe in innocence. Those who believe in sin do not believe in health. Those who believe in sin are busy finding ways to punish themselves. Sickness becomes equated with, and actually seen as "guilt", when the Atonement is accepted and innocence is recognized as true.
When somenone is sick, (and in ACIM terms, sicknes means any kind of sense of separation from God, not just physical expressions of sickness), when they are mentally separated from truth, they are believing in guilt. And this will manifest as forms of sickness in the body and in other areas of life. And these will be areas of guilt and punishment.
If you are willing to accept salvation for yourself, which means to become able to ACCEPT that the Atonement is true, ie there is no such thing as real sin and real sin has not happened, your clarity and willingness then allows you to claim "for others" that they too are worthy of love because they too have not sinned.
Their sins become more obvious as being untrue of them, as much as your own are untrue of you. You will not see their sickness as truth or as reality. You will recognize the person is experiencing guilt and is acting against themselves. They do not deserve this. They deserve the same love that you have come to recognize you deserve, because of your acceptance of the Atonement's message and reality.
Cancelling the illusion of sickness, masquerading as sickness, seeing past it to recognize the guilt, and recognizing it as untrue, gives you the power to set your brother free. This is why Jesus was able to tell people that their sins were forgiven. It wasn't that they had been pardoned, but that He knew their sins were not really sins. Sin is not real.
He instead recognized their state of believing they were sinful needed to be healed and undone. He was so certain and clear that they were nothing short of completely innocent, that He did not get hung up on their illusion of sickness for a second. And that gave him the ability to dispel those illusions and cast them off like they were nothing.
If the Atonement is true for you, it MUST be true for your brothers and sisters also. The Atonement tells you that there is no division and there is no separation and there is no set of rules that apply differently to one person and not another. If love applies to you, because you are forgiven, then it also applies to everyone around you. They are seen as lovable because you see yourself as lovable.
Because they are you. They literally are your shared identity in Christ. One.