This is the 4582nd most frequent Turkish word.
başlayacağını
"Başlayacağını" means "that [someone/something] will start."
In this sentence, 'başlayacağını' indicates the subject's intention or plan to commence an activity in the future, expressed in third person singular form.
Yarın sabah erkenden bir maratona başlayacağını söyledi.
She said that she would start a marathon early tomorrow morning.
Here, 'başlayacağını' denotes the scheduled future occurrence of an event, namely, the reopening of schools.
Okulların gelecek ay başlayacağını duyduk.
We heard that schools would start next month.
In this usage, 'başlayacağını' reflects the subject's expressed plan to initiate a specific action.