The word desvelaban is the imperfect form of desvelar in the third person plural.
See the full desvelar conjugation

desvelar

Y que decoró con esfuerzo y trabajo los sueños que lo desvelaban cada noche.
And after much work and effort, the dreams which kept him awake at night came to life.
Los avances desvelaban el final del filme.
The trailer spoiled the end of the movie.
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