Learn French: Elle m'a expliqué qu'elle était arrivée la veille. - She explained to me that she had arrived the day before.

In reported speech, when the main verb (a expliqué) is in the past, an event that happened even earlier shifts to the pluperfect tense (le plus-que-parfait). If it were direct speech, she would have said Je suis arrivée, which becomes elle était arrivée to show it happened before she explained it.

Why is the phrase for the day before translated as la veille instead of hier?

This is a classic rule of French reported speech. When the main verb is in the past, relative time words shift to fit the narrative timeframe. Direct speech uses hier for yesterday, which must change to la veille for the day before in indirect speech.

Why is there an extra e at the end of arrivée?

The verb arriver takes être as its auxiliary verb. Because of this, its past participle must agree with the subject. Since the subject elle is feminine singular, we add an e to make it arrivée.

What would it mean if we used the imparfait (elle arrivait) instead of était arrivée here?

It would change the timeline. Using arrivait would mean she was in the process of arriving at the exact same time she was explaining it. Using était arrivée makes it clear her arrival was a completed action that happened before the explanation.