Learn French: La moitié de la troupe se sera repliée. - Half of the troop must have retreated.

This is the epistemic use of the future perfect tense. In French, you can use this tense to express a strong probability, deduction, or assumption about something that happened in the past. It is the direct equivalent of saying "must have" or "probably" in English.

Why does the verb use sera (from être) instead of aura (from avoir)?

The verb used here is se replier (to retreat or fall back), which is a pronominal verb. In French, all pronominal verbs use être as their auxiliary verb in compound tenses, including this one.

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

Because the auxiliary verb is être, the past participle repliée must agree in gender and number with the subject. The subject here is La moitié de la troupe (feminine singular), so we add an e.

If the subject were "half of the soldiers" (la moitié des soldats), would the verb still be singular?

Usually, no. With fractions and quantities like la moitié de (half of), the verb typically agrees with the noun that follows it. Since la troupe is singular, the verb is singular. If it were plural (des soldats), you would use the plural: se seront repliés.