Death Throes does not happen if you drop to 0 hit points
If you revert as a result of dropping to 0 hit points, any excess damage carries over to your normal form.
In other words, when you drop to 0 hit points you don't die, instead you directly revert into your normal form, so Death Throes does not take effect.
Death Throes might happen, if you die while shapechange is ongoing.
You can die while shapechange is ongoing in a number of ways, for example if you are targeted by power word kill or if you gain 6 levels of exhaustion.
However, the rules for Concentration state:
Some spells require you to maintain concentration in order to keep their magic active. If you lose concentration, such a spell ends.
- [...] You lose concentration on a spell if you are incapacitated or if you die.
Hence, if you die while shapechange is ongoing, you simultaneously revert and use Death Throes, so it's unclear what should be resolved first. If Death Throes is resolved first, you explode and then your scattered corpse reverts. If reverting is resolved first, then Death Throes can no longer happen.
Xanathar's Guide to Everything (p. 77) has an optional rule for simultaneous effects:
Most effects in the game happen in succession, following an order set by the rules or the DM. In rare cases, effects can happen at the same time, especially at the start or end of a creature’s turn. If two or more things happen at the same time on a character or monster’s turn, the person at the game table — whether player or DM — who controls that creature decides the order in which those things happen. For example, if two effects occur at the end of a player character’s turn, the player decides which of the two effects happens first.