What does the Divining Shells ability do ? What does kill this, tap mean to trigger the ability ?

Yes — you read the cost right: "Kill this, [T]" means you exhaust (tap) Divining Shells and then kill it, so the gear is destroyed as part of paying for the ability. [T] is just the old shorthand for the exhaust symbol (current rules shorthand is [E], CR 135.2.e.2).

The full ability is "[Action] — Kill this, [T]: Give a unit +2 [M] this turn," so in exchange for tapping and destroying the gear, one unit gets +2 Might until end of turn. The +2 sticks even though the gear is gone, because it's the ability resolving, not the gear's continuous effect. Both costs must be payable: an already-exhausted Divining Shells can't be used, since you can't exhaust an exhausted permanent (CR 415.4).

Its other line, Vision, triggers when you play it: look at the top card of your Main Deck and you may recycle it. Note the [Action] on the ability grants timing permission only — it can also be activated during showdowns on either player's turn — but it's still a one-shot, because the gear kills itself to pay for it.