It is working with Brands (which are self-casting - a brand is not a separate entity from you any more than a fireball is), and the only minion spell it can affect. It affects your skills whether used by yourself or by totems/traps/mines.
The only minion skill it can interact with is Absolution, since that is the only minion skill that is both elemental and deals damage. Damage you deal with that skill will have 100% chance to deal Triple Damage. This does not affect the minions, because their damage is not damage you're dealing with an elemental spell.
Summon Raging Spirits, for example, is not a skill that deals any damage with at all, so there's nothing to triple - it just creates a minion (which does not have your staff) and that minion uses melee attacks to deal it's own damage, independant of you or the spell that created it.