To answer the question of how this is used, you need definitions for two things: activated abilities and mana abilities.
602.1. Activated abilities have a cost and an effect. They are written as "[Cost]: [Effect.] [Activation instructions (if any).]"
406.1. A mana ability is either (a) an activated ability without a target that could put mana into a player’s mana pool when it resolves or (b) a triggered ability without a target that triggers from a mana ability and could produce additional mana. A mana ability can generate other effects at the same time it produces mana.
"a" being the only one that matters in this example
That means, that illusionist bracers will give you double the effect, whenever you use the equipped creature's ability, if the ability is written as: "[Cost]: [Effect.] [Activation instructions (if any).]" and if the ability isn't a mana ability.
It doesn't work on LLanowar Elves, since it's ability is a mana ability.
It does work on Deathrite Shaman (for all abilities), since they target and add mana it's not a mana ability.
It doesn't work on Solemn Simulacrum, since it doesn't have any activated abilities. It has triggered abilities, but those are not written in COST : EFFECT form.
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