The first time Isaac takes damage while in an uncleared room, time is stopped for 2 seconds and Isaac becomes a ghost and leaves his body behind, which drops to the floor. While in ghost form, Isaac has flight and spectral tears and the next hit he takes is negated.
Isaac's tears and speed are greatly increased while time is stopped, although this is not shown on the HUD.
Isaac's ghost re-enters his body after the room is cleared or if he gets hit again.
Notes
This item has a 1% chance to drop when defeating Clutch.
The damage blocked by being in ghost form also affects self-damage and red-heart draining sources, like Devil Beggars, IV Bag, etc.
Receiving lethal damage can potentially save Isaac's life if he manages to heal before the ghost reenters the body.
Dull Razor may be used to safely activate Astral Projection as The Lost.
It's also possible to leave the room while the ghost is outside the body, either by bombing it or having items that keep doors open, such as Door Stop or Mercurius. This grants Isaac an additional chance to heal.
Teleporting outside of the room will not work.
Astral Projection cannot be triggered inside of Boss Rush.
Synergies
Hypercoagulation: Due to the effective speed up, the heart used to activate Astral Projection can easily be collected before it disappears.
IV Bag / Dull Razor / Other sources of self damage: If used in a hostile room, Astral Projection will trigger, effectively granting flight and Holy Mantle for the room.
Old Bandage / Gimpy: When getting hit with only one heart left, Astral Projection allows for picking up the dropped heart in the brief time of invincibility even when the player has no hearts left. Failing to do so or having no hearts dropped results in guaranteed death however, but with luck this loop can be used to potentially save runs.
Interactions
Holy Mantle: The shield will activate before Astral Projection.
The Lost / Tainted Lost: The Lost will continue to fight after taking damage, however he will instantly die upon taking damage again or once the room is cleared.
If The Lost receives at least half a Soul Heart through any means while the effect is active, he will survive. Black Hearts cannot be used for this.
Plan C / Other insta-kill effects: Will kill Isaac without triggering the ghost effect.
Rock Bottom: Does not preserve the tears and speed bonus Isaac receives after activating Astral Projection.
Spirit Shackles: This item will not activate if Isaac received lethal damage when Astral Projection activated.
This item has a 1% chance to drop when defeating 
Clutch.
Devil Beggars, 
IV Bag, etc.
Dull Razor may be used to safely activate Astral Projection as The Lost.
Door Stop or 
Mercurius. This grants Isaac an additional chance to heal.
Boss Rush.
Hypercoagulation: Due to the effective speed up, the heart used to activate Astral Projection can easily be collected before it disappears.
Holy Mantle for the room.
Old Bandage / 
Gimpy: When getting hit with only one heart left, Astral Projection allows for picking up the dropped heart in the brief time of invincibility even when the player has no hearts left. Failing to do so or having no hearts dropped results in guaranteed death however, but with luck this loop can be used to potentially save runs.
The Lost / 
Tainted Lost: The Lost will continue to fight after taking damage, however he will instantly die upon taking damage again or once the room is cleared.
Soul Heart through any means while the effect is active, he will survive. 
Black Hearts cannot be used for this.
Plan C / Other insta-kill effects: Will kill Isaac without triggering the ghost effect.
Rock Bottom: Does not preserve the tears and speed bonus Isaac receives after activating Astral Projection.
Spirit Shackles: This item will not activate if Isaac received lethal damage when Astral Projection activated.
Guillotine.
