The security guard admitted that he had fallen asleep at his desk. By the time he woke up, the thieves had already stolen the priceless painting.

The past perfect (had + past participle) is used to show that an action was completed before another event in the past. The guard falling asleep and the thieves stealing the painting both happened before the guard woke up and admitted his mistake.

The adverb "already" is a strong hint that the action was fully completed before the second past action (waking up) occurred.