Learn Spanish (Spain): Dijo que volvería al día siguiente. - He said he would come back the next day.

This is an example of reported speech. The original speaker probably said 'I will return' using the future tense. Since we are reporting this statement after the fact (Dijo que...), the future tense shifts to the conditional tense (volvería), which translates perfectly to English as 'would return'.

Why does the sentence use al día siguiente instead of mañana?

Because the reporting verb (Dijo) is in the past, the time reference changes. If he originally said 'I will return tomorrow' a week ago, 'tomorrow' has already passed. Just like in English, mañana (tomorrow) shifts to al día siguiente (the next day) when reporting speech in the past.

Why do we use al before día siguiente?

In Spanish, to say 'on the next day', you use the preposition a plus the masculine article el, which combine into the single word al. You cannot just use el día siguiente on its own when it acts as an adverbial time phrase telling us when an action happened.

How do we know who is coming back since there are no pronouns in the sentence?

The verb dijo means 'he/she/you said'. The verb volvería can mean 'I would return' or 'he/she/you would return'. Without pronouns like él or yo, Spanish relies on context. We naturally assume the person who spoke is also the person returning. If he meant someone else, a pronoun would have been added to clear up the confusion.