The moons are just there, there's no precalculated purpose. It's left-over rubble that didn't make it to form a planet. Or it was a planet but now destroyed. Or it was part of another solar system, but no longer there.

There are many aspects that makes a planet livable. Jupiter's atmosphere is not one of them. Most is gaseous.

Our moon was formed by the rubble of a collision with another planet. This collision itself made things surface from the planetary core that would not be there without that collision, like Iron. It is likely that it also caused the tilt creating seasonal variation, while the Moon itself is causing a slightly more frequent variation in the form of tides and perhaps affecting tectonic plates. All these variations and stirring of material may be the cause of live.