Succulent plants store water in their tissue. That, by definition, is what makes it a succulent plant. Succulents use the water they store in their tissue to help them survive long dry periods. Since cacti and other succulents, as a general rule, grow in dry places, they take up water through their roots very quickly and efficiently. On the other hand, they cannot quickly get rid of excess water. While this all works out fine in the wild, people can supply more water than the succulents' needs when succulents are grown in cultivation. This is actually a typical situation, and the plants in cultivation look plump and happy compared to their habitat counterparts.
But the real problems start when people do not just give their succulent more water than it needs, but more water than it can handle. This is overwatering. When a succulent is overwatered, it will swell up more and more, and often the stem will become so saturated that it splits open in one or more places. While a succulent can usually recover from this, the other result of overwatering, especially when combined with cold temperatures, is rot. Sometimes just the roots will rot off, but often the whole plant will turn to mush.
The first thing to do with an overwatered succulent is to stop watering it. In most cases, repotting would also be a good idea. When the plant is unpotted, you can let it stay unpotted to dry out for a week or two and then repot it. If the plant body has split in any place, the damage is permanent, and there is nothing you can do but let it scar over. If the plant has started to rot, saving the plant will depend on how far the rot has set in.