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I was looking at the Transcendence, Platinum Angel interaction. As it is explained in a few places online

Transcendence has what is called a 'State Trigger'. It triggers whenever the condition is met provided there isn't a copy of that trigger on the stack already. What will happen is Transcendence will continue to trigger over and over again since you don't lose the game due to Platinum Angel. There is no way to stop this without a player doing something and that is not required so the game ends in a draw.

But why does this not happen if the player reaches negative life? I know from playing MTGO a player can continue playing the game with negative life if the Platinum Angel is in play.

119.6. If a player has 0 or less life, that player loses the game as a state-based action. See rule 704.

What prevents the death trigger from firing over and over similar to Transcendence? When is loss checked if Platinum Angel leaves the field if it's not done continuously?

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The normal loss condition of having zero or less life happens only once per check of state-based actions. The Transcendence loss ability triggers continuously in an infinite loop, with no way to continue playing.

The loss to zero or less life happens due to state-based actions. They are checked often, but not infinitely often. They are checked whenever a player would gain priority, which happens only at specific points during any given turn.

117.3a The active player receives priority at the beginning of most steps and phases, after any turn-based actions (such as drawing a card during the draw step; see rule 703) have been dealt with and abilities that trigger at the beginning of that phase or step have been put on the stack. No player receives priority during the untap step. Players usually don’t get priority during the cleanup step (see rule 514.3).

117.3b The active player receives priority after a spell or ability (other than a mana ability) resolves.

117.3c If a player has priority when they cast a spell, activate an ability, or take a special action, that player receives priority afterward.

117.3d If a player has priority and chooses not to take any actions, that player passes. If any mana is in that player’s mana pool, they announce what mana is there. Then the next player in turn order receives priority.

117.5. Each time a player would get priority, the game first performs all applicable state-based actions as a single event [..]

Platinum Angel prevents that single loss event each time a player would gain priority, and that's that.

The loss ability of Transcendence, however, has a state trigger. It will trigger whenever the game state in question has been reached and no instance of that ability is currently on the stack. As soon as the trigger leaves the stack (or doesn't enter it) for any reason, and the game is still in the necessary state, the ability will trigger again right away, without anyone being able to react. Platinum Angel's ability removes the Transcendence trigger from the stack by preventing it. The Transcendence ability never enters the stack, it immediately triggers again, and no player will gain priority, resulting in a draw.

603.8. Some triggered abilities trigger when a game state (such as a player controlling no permanents of a particular card type) is true, rather than triggering when an event occurs. These abilities trigger as soon as the game state matches the condition. They’ll go onto the stack at the next available opportunity. These are called state triggers. (Note that state triggers aren’t the same as state-based actions.) A state-triggered ability doesn’t trigger again until the ability has resolved, has been countered, or has otherwise left the stack. Then, if the object with the ability is still in the same zone and the game state still matches its trigger condition, the ability will trigger again.

614.6. If an event is replaced, it never happens. [..]

603.3. Once an ability has triggered, its controller puts it on the stack as an object that’s not a card the next time a player would receive priority.

So what happens is, Transcendence triggers. The next time a player would gain priority, the Transcendence trigger wants to enter the stack, but Platinum Angel prevents that. Transcendence immediately triggers again, Platinum Angel prevents it again, and so on, before the player can ever gain priority. Therefore, it's a draw because there is a loop with only mandatory actions.

721.4. If a loop contains only mandatory actions, the game is a draw.

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    “The Transcendence ability never enters the stack” — this is false. It is put on the stack, then players get priority as usual, then it resolves, however, the angel causes it not to have any effect. Any player, if they wish, can use the priority to cast a removal spell on the angel or do something else. Only if no-one removes the angel, the loop results in a draw. – Roman Odaisky Mar 7 at 20:21
  • What is this, a badly written computer program? If the trigger doesn't advance state, it can be shortcutted out. – Joshua Mar 8 at 20:39
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In addition to Hacksworth's good answer, here are details on the way that state-based actions are checked:

704.3. Whenever a player would get priority (see rule 117, “Timing and Priority”), the game checks for any of the listed conditions for state-based actions, then performs all applicable state-based actions simultaneously as a single event. If any state-based actions are performed as a result of a check, the check is repeated; otherwise all triggered abilities that are waiting to be put on the stack are put on the stack, then the check is repeated. Once no more state-based actions have been performed as the result of a check and no triggered abilities are waiting to be put on the stack, the appropriate player gets priority. This process also occurs during the cleanup step (see rule 514), except that if no state-based actions are performed as the result of the step’s first check and no triggered abilities are waiting to be put on the stack, then no player gets priority and the step ends.

And the first state-based action:

704.5. The state-based actions are as follows: 704.5a If a player has 0 or less life, that player loses the game.

So the state-based action checks are repeated, but only if an action actually happened as a result of the check. Because Platinum Angel prevents 704.5a from actually happening, then there is no action that happened (assuming no other applicable actions happened at the same time), and so the check is not repeated. They will be checked again next time a player is about to receive priority, but not until then.

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