Activated abilities, legend abilities, and "use"

An activated ability is a repeatable effect with a cost, written "cost: effect" (rule 377 and rule 377.1). You decide when to use it, it goes on the chain like a card, and the opponent may respond. The only hard part is timing: the same Open, Closed and Showdown checks as for cards, plus any condition printed on the ability.

Worked example. Your legend is Lee Sin, Blind Monk: "1 Energy, Exhaust: Buff a friendly unit". It has no Action or Reaction, so you can activate it only in your own Main Phase while no chain exists and no showdown is in progress (rule 381 and rule 381.1). In a showdown, or with your opponent's spell on the chain, you cannot. Now compare Kai'Sa, Daughter of the Void, whose legend ability is "Exhaust: Reaction, Add one rainbow rune. Use only to play spells." That Reaction lets Kai'Sa's controller activate it even in a Closed State on the opponent's turn (rule 813.1.c.2), say to pay for a Reaction spell in response. Because it Adds resources, it resolves immediately and cannot be responded to (rule 337.2).

Who and when. The player who controls the source decides when and whether to activate it (rule 378). If the source is uncontrolled the owner decides (rule 378.1), and a few abilities name other players who may use them (rule 378.2). Legends can have activated abilities (rule 174.8). By default an activated ability can be activated only on the appropriate player's turn and during an Open State, which is its controller's Main Phase (rule 381 and rule 381.1). Keywords widen that. A Showdown State allows only abilities with Action or Reaction (rule 308.1.a), and a Closed State only Reaction (rule 309.1.a). On an activated ability Action means "this can be activated during showdowns on any player's turn" (rule 806.1.c.2). Reaction adds Closed States and includes everything Action allows (rule 813.1.c.2 and rule 813.1.b). Neither takes away the default timing (rule 806.2 and rule 813.2).

The chain. An activated ability goes on the chain with no card to represent it, so players need to remember that the turn is now Closed (rule 377.3 and rule 377.3.a.1). Opponents can respond as if a card had been played (rule 377.3.b.2).

Costs and conditions. The Exhaust symbol means "Exhaust this" (rule 415.5). A game object that is already exhausted cannot be exhausted again (rule 415.1.b), so an Exhaust cost cannot be paid by an exhausted source (rule 415.4). Card text says "use" or "play" for activating (rule 377.2). A condition on using an ability, such as Xerath's "use this ability only while I'm at a battlefield", must be true to activate it at all (rule 377.2.b). If using an ability is part of a trigger condition, that condition is fulfilled when the ability resolves, not when it is activated (rule 377.2.a).

A different "activate". When an effect tells a player to activate a kind of triggered ability, such as Reckoner's Arena activating conquer effects when you hold, that player checks each named effect's condition as if the named part were fulfilled and puts it on the chain only if every condition is met (rule 383.4.g and rule 383.4.g.1). The named event does not really occur, so effects that add extra triggers of it do not apply (rule 383.4.g.2).

Common mistakes. "It's my turn, so I can use it": not in a showdown or with a chain present unless it has Action or Reaction. "Action lets me respond": Action adds showdowns to the default timing and nothing more, and a chain closes the turn to everything but Reaction.