Activating the ability granted to the enchanted creature causes a "regeneration shield" to be created for it. The next time that creature would be destroyed that turn, the regeneration shield is used up instead. This works only if the creature is dealt lethal damage, dealt damage from a source with deathtouch, or affected by a spell or ability that says to "destroy" it. Other effects that cause the creature to be put into the graveyard (such as reducing its toughness to 0 or sacrificing it) don't destroy it, so regeneration won't save it. If it hasn't been used, the regeneration shield goes away as the turn ends.
To work, the regeneration shield must be created before the enchanted creature is destroyed. This usually means activating its ability during the declare blockers step, or in response to a spell or ability that would destroy it.
If the enchanted creature is dealt lethal damage and is dealt damage by a source with deathtouch during the same combat damage step, a single regeneration shield will save it.
Trollhide needs to be attached to the creature as the ability that creates the regeneration shield is activated, but after that, the regeneration shield will be created and remain in effect even if Trollhide is no longer attached to the creature.