The phrase “Aussitôt que” means “as soon as,” and it introduces a past action that happened immediately before another past action.
The second clause “je leur ai dit” is in the passé composé tense.
Therefore, the action of knowing the truth must have happened before I said it — requiring an earlier past tense.
In literary and formal French, when two consecutive past actions are mentioned, the earlier one often uses passé simple.
The passé simple of “savoir” (to know) for the first person singular “je” is sus.
So the completed sentence is: Aussitôt que je sus la vérité, je leur ai dit.