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

There's a variety of perspectives about the use of terminology such as "Jesus took our illnesses upon himself on the cross" (from perhaps "sickness is something that shouldn't affect Christians" to perhaps a focus on the full-application being in the future kingdom) that this statement could cover. What are the background texts to support such a statement (one would not have to agree with the exegesis to present the texts used to support the doctrine). One commonly referenced text would be Isa 53:4-5 which concludes > and with his wounds we are healed What is the range of ideas that "Jesus took our illnesses" covers and what key texts are used to build up that argument.