The word desayunaban is the imperfect form of desayunar in the third person plural.
See the full desayunar conjugation

desayunar

Con frecuencia se veían solo por la mañana, cuando desayunaban.
They saw each other only in the morning when she served him breakfast.
Habían organizado una pantalla grande en la sala de estar. Y cada mañana desayunaban con su abuela.
They had set up a big screen in their living room, and every morning they had breakfast with their grandmother.
Desde entonces se encontraban cada mediodía en el muelle, desayunaban juntos, cenaban, paseaban y admiraban el mar.
Since then, they met up every afternoon at the pier, had breakfast together, had dinner, took walks and admired the sea.
¿A qué hora desayunaban durante las vacaciones? - Desayunamos a las ocho todos los días.
What time did you have breakfast while you were on vacation? - We had breakfast at eight o'clock every day.
Bueno, tengo que encontrar el auto de esta mujer así que cualquier cosa que tú o Larry puedan hacer, lo apreciaría. Entonces la última vez que vio a su esposa e hija fue cuando desayunaban.
Well, I got to find this woman's car so anything you or larry can do I mean, I would appreciate it so the last time you saw your wife and daughter they were eating breakfast
Según un matrimonio mayor: los dos desayunaban el día de sus bodas de oro. La mujer untó mantequilla en la corteza del pan y se la ofreció al marido, quedándose ella con la miga.
According to an older couple: The two are having breakfast on their Golden Anniversary. The woman spreads butter on the crispy crust of toast and hands it to her husband, keeping the soft dough to herself.
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