Phasing out doesn't cause any “leaves the battlefield” abilities to trigger. Similarly, phasing in won't cause any “enters the battlefield” abilities to trigger.

Choices made for permanents as they entered the battlefield are remembered when they phase in.

Notably, this means you cannot equip Robe of Stars to another creature while it is phased out. If you activate the Astral Projection ability of an attached Robe of Stars in response to the equip ability, Robe of Stars will be phased out before the equip ability resolves and will not be moved.

An attacking or blocking creature that phases out is removed from combat.

Permanents phase back in during their controller's untap step, immediately before that player untaps their permanents. Creatures that phase in this way are able to attack and pay a cost of during that turn. If a permanent had counters on it when it phased out, it will have those counters when it phases back in.

Phased out permanents are treated as though they don't exist. They can't be the target of spells or abilities, their static abilities have no effect on the game, their triggered abilities can't trigger, they can't attack or block, and so on.

Any continuous effects with a “for as long as” duration, such as that of Mind Flayer, ignore phased-out objects. If ignoring those objects causes the effect's conditions to no longer be met, the duration will expire.

As a creature is phased out, Auras and Equipment attached to it (including Robe of Stars) also phase out at the same time. Those Auras and Equipment will phase in attached to the creature they were attached to when they phased out.

If an opponent gains control of one of your creatures, that creature phases out, and the duration of the control-change effect expires before it phases back in, that creature phases in under your control as that opponent's next untap step begins. If they leave the game before their next untap step, it phases in as the next untap step begins after their turn would have begun.