What do Christians mean by "Jesus took our illnesses upon himself on the cross" and why?

I am afraid you are wrong Mr. Affable geek on what sickness means in Isaiah 53:4 . If you study it out in the Hebrew, you will see that it means literally sickness ! Also Matthew 8:17 shows Jesus healing the sick , to fulfill Isaiah 53 ! And let me say that Jesus carried away OUR sicknesses. not just those people in Matthew 8:17 . Because you see , I and you are an " OUR " ! Do you see that ? Also Psalms 101:4 says that He forgives all OUR sins and heals all OUR diseases. Jesus not only carried away our sins. But He also carried away the consequences of our sins , namely those things under the curses of those who disobey the ten commandments, one of those curses are SICKNESS ! But thank God that Jesus became a curse for us and redeemed us from the curse ! We do not have redeemed bodies now . But Jesus patches up THESE bodies for us until we get our glorified bodies. Our new bodies will never need patching up. So in the meantime , if you are sick, know that He forgives all your sins and heals all your diseases. This is why James said, Is ANY among you sick ? call for the elders, and the prayer of faith WILL heal the sick ! So for the sake of all the readers out there, make sure you handle the word of God accurately !

Healing is part of the 'atonement', Christ's redemptive work on the cross.

Numerous supporting scriptures also attest. He heals all our diseases, Psalms 103:3. By His stripes we were healed, Matthew 18:7 & Isaiah 53. He bore our illnesses, 2 Peter 2:24. None of the diseases of Egypt, Deut 28:60.

The issue of healing is a rift in the body. While some claim all miraculous sign gifts have ceased, others claim to have witnessed and even participated in them. The difference between dialectical argument and experience gets lost in the shuffle, and despite the availability of videos, testimonies, and real life encounters, people are not obliged to believe (by them).

While some claim a partial and some a complete cessation of the miracles and healings demonstrated in the life of Jesus through the written Gospels, other testimonies throughout history have demonstrated supernatural endeavors (such as the Catholic church, which only beatifies those who have some level of miracles). Even today, while some would doubt, some ministries claim to have upwards of 500 people literally raised from the dead in the past few decades, in the midst of their evangelistic and missions work to the poor of Southern Africa and parts of Mexico.

Indeed, healing tents only several decades ago dotted America in the 1950s, as a wave of healing evangelists went across the nation. Excess and moral failure ultimately caused it's collapse, bringing reproach in general to the subject in the current day.

Theologically, however, healing is one of the hallmarks of Christianity, at lest in some circles. Christ came and demonstrated it, taught faith, and said anythibg was possible if we believed. Men such as Kenneth Hagin took Him at His Word, and is reported to have witnessed many uncounted miracles in His ministry.

As to why some might not get healed, therr are a variety of reasons, other than simply the catch phrase, "Uou don't have enough faith, brother". Indeed, Paul mentioned a few practical reasons for sickness and early death in the Corinthian church in his first epistle.

But, regardlesd, while some would disagree with healing being normative for today, others wholeheartedly day it is. And, as for who gets healed and who does not, usually, the best answer is, thank God for what He is doing.

In life, when you have a problem you have a cause for action. You talk to God about it—that, is prayer. And there, you have your causal relationship between prayer and life.

Your problem could be a health issue that won't go away. You are stuck with a wrong combination of doctors, treatments and circumstances. It is beyond you.

Where fact reaches its limits, faith steps in. You turn from the science of the cure to the promise of a cure. A woman, who for twelve years had suffered much at the hands of doctors and treatments, now was faced with the situation wherein her circumstance had put medical intervention beyond her reach. She turned to Jesus because she now believed that He only could cure her. By faith she reached out and touched His garment. That was her prayer. Validation for her faith may be found in the promises Jesus had publicly made, for instance, Matthew 21:22 and John 15:7. Confirmation for what Jesus was claiming came from the writings of the prophets of old, particularly Isaiah 53:4-5. The woman, convinced that Jesus' office and promise covered her condition, went ahead and took action. She bought her prayer to Him, He responded.

He took her disease and healed her infirmity, bearing in Himself (empathizing with her) her pain and suffering. Read the story in Luke 8:43-48.

Isaiah 53 is not just about sin. Much more so, it is certainly not just about disease.

It is a picture of Christ on the cross, so all history leads up to this and all the future feeds back to this. The whole 'curse' of Adam was placed upo Him, and from Him the whole 'blessing' of Abraham is made. This is nothing less than the fall and return to paradise. So in a sense all salvation is here in its primitive kernel.

Even Rabbinic sources before Christ took the Messiah to undergo physical suffering and that would include the healing of physical diseases, not just sin. This can be seen from the Talmud identifyng th Messiah as the 'leperous one'.

Babylonian Talmud (Sanhedrin 98b): "Messiah ...what is his name? The Rabbis say,' The leprous one'... for it says, 'Surely he has borne our sicknesses' etc. (Isa.53,4)."

From the New Testament view:

16When evening came, many who were demon-possessed were brought to him, and he drove out the spirits with a word and healed all the sick. 17This was to fulfill what was spoken through the prophet Isaiah: “He took up our infirmities and carried our diseases.” (Mathew 8:16-17)

This makes it sufficiently clear that these words include demon possession and illnesses of body.

24He himself bore our sins in his body on the tree, so that we might die to sins and live for righteousness; by his wounds you have been healed. 25For you were like sheep going astray, but now you have returned to the Shepherd and Overseer of your souls. (1 Peter 2:24)

This also makes it clear that this means he would take away our sins.

So the idea seems to be that that Jesus is the great Messiah, the Healer, Comforter, Sin Bearer and killer of all; sin, hell, sorrow, death, and sickness, through his atonement. This does not mean all sickness and death is removed now, for salvation in history had developments and the full achievement of Christ’s work is in the future. All sorrow and suffering and sickness will not be removed until his future return to judge the living and the dead. However, when God met mankind, in the God-Man, it was impossible for the bizarre situation not to drive all sickness and demon possession away. The same will be true when he returns in glory:

16Never again will they hunger; never again will they thirst. The sun will not beat upon them, nor any scorching heat. 17 For the Lamb at the center of the throne will be their shepherd; he will lead them to springs of living water. And God will wipe away every tear from their eyes.” (Rev 7:16-17)

This is the final dimension of Isaiah 53.

In Isa 53, the word you are translating "illnesses" is more often translated as "transgressions" or "iniquities." Both of those words highlight the fact that Jesus took our sins with him on the cross, and by the wounds he suffered, we were healed - from the iniquity of sin.

The basic idea of salvation in Christainity is often summarized as this -

1. We are sinners

2. Because of this sin, we deserve eternal death - separation from God

3. Jesus paid the penalty for that sin by dying on the cross, thus we are healed and given eternal life.

(For the purposes of this answer, lets not get into all the different ideas of atonement, etc... I'm going for the overview here!)

The point is "illness" is not a physical thing here, but a legal one. (Indeed, I've never seen that translation before - perhaps its just being confused with iniquities? An iniquity is an evil, a cause of suffering and distress, not necessarily a bodily thing.) We are "suffering" from sin, and the Suffering Servant of Isaiah 53 pays for that sin, so we are healed.

Beyond that, I've never seen a promise that Christians would never get sick. Even the early church recognized that believers would still die. The Good News, however, is that in our resurrected state, there would be no more corruption in our bodies - i.e. no sickness - but that's on the other side of the Jordan, not this one!