How exactly does stored water (for a survival situation) go "bad"?

You're pretty much correct. If your stored water came from a safe drinking source, treating stored water for possible biological growth will typically render it safe again. If you live in an area with a treated municipal water supply and reasonably good source water, it usually will store safely for a very long time as long as you are storing it in sterile, sealed containers that do not leach chemicals/metals into the water.