message: What do Mormons mean when they claim that Jesus is their savior? Exactly what do they do when they make Jesus their savior?
To understand what Mormons mean by the word "salvation," you have to understand their word "exaltation."
For Mormons, there are three heavens; all but the very most vile murdering sinners go to heaven no matter what they believe. They go to the Telestial Kingdom, (which one Momon Mormon authority described as so beautiful you might kill yourself to go there). The other two kingdoms are the Terrestrial and the Celestial. The Celestial kingdom is divided into three levels, the higest of which corresponds to what Bible-believing Christians call heaven. Mormons get there only through rigorous obedience to numerous laws.
In Mormon theolgy, the work of Christ's Atonement merely assures that all men will be resurrected after they die. That is the salvation won through accepting Jesus' work on the Cross. When they say they are saved, that is what they refer to.