If you read the general context of this passage in Chorinthians you will find that there is a lot of stuff about church order, roles in the church, how to deal with problems inside and outside the body of believers, etc.
The verse you quote in particular is starts to make sense when you see it as a reference to to church discipline and understand what the purpose of that discipline is. In this case we're specifically talking about the last stage of possible disipline: excommunication. This is the final step that a church will take in trying to bring the life of somebody who claims to be a believer into alignment with their confession. Although the step is a drastic one, the purpose is no different than the earlier steps: Christ receiving glory as people people are saved and brought from darkness to light.
Excommunication (or total removal of a person from the church) serves two purposes.
It protects the church. The church is supposed to be holy: the bride of Christ spotless and reserved for Him. Of course the church is also comprised of sinners, but we are in the process of being made holy. Step by step we are supposed to be conforming to his standards and looking more like him. If somebody is moving the oposite direction — deliberatly and intentionaly continuing to sin and justify their sin; then removing them from the body helps stop the corruption from spreading.
It stands as a severe warning. The hope is that by being rejected by the church they will be made aware of the seriousness of their sin before God and realize that they need to repent. Removal from the church only happens after teaching, warning, and internal discipline measures have failed — in other words the person is not listening to lighter warnings. They still insist they are Christian but are also insistent on living in a manner inconsistent with Christianity. By rejecting their claim to "be Christian" and not allowing them to participate in church functions, they are forced to consider whether they really are Christian.
Back to your verse in particular: "Deliver this man to Satan" is a reference to excommunication -- where the church rejects a persons claim to be Christian and forces them to live outside the community of the church. If the person truly has been chosen and is being saved by the Lord, finding themselves outside the blessing of the church is an opportunity for them to discover and destroy another part of their flesh: repenting of their worldly deeds and re-dedicating themselves in alignment with their stated belief.
If they do so, they are to be welcomed back into the church and the whole process will have been for their well being (and a step in their eventual complete salvation). If not, the process will have proven that they never did belong to Christ and actually serve another master.
If you turn the whole problem inside out it becomes more apparent why this works. The alternative is for the church to turn a blind eye to serious sin — allowing people who profess to be followers of Christ to live like followers of the devil as if there were no consequences. Doing so would produce a false sense of temporary security and they would find out on the final day of Judgement that what Jesus had in mind for his true followers was something else entirely.
By enacting current temporal consequences for non-repentant sin on this earth, the church grows in ito it's roll of a spotless bride.