Learn Farsi: من به ایشان عرض کردم که شما خسته هستید.

While ishân is the 3rd person plural pronoun ("they"), in formal and polite Persian it is heavily used as a respectful pronoun for a single third person ("he" or "she"). Here, the speaker is showing esteem for the person they were talking to, so they choose ishân instead of the everyday singular pronoun u.

Since the English translation says "you are tired" (singular), why does the Persian sentence use shomâ and the plural verb hastid?

This is the core of the Persian polite address system. To show respect to a single addressee, you use the plural pronoun shomâ and match it with a 2nd person plural verb (like hastid). If the speaker were talking to a close friend, a family member, or a child, they would use the informal singular pronoun to and the singular verb hasti.

Why use the verb 'arz kardam instead of the standard verb for "I said" (goftam)?

This is a great example of Persian politeness (ta'arof). When interacting with someone you respect, it is customary to "elevate" them and "lower" yourself. 'Arz kardan literally means "to submit a petition" or "to humbly state." By using this humble verb for your own action instead of the standard goftam, you show deference to the person you are speaking to.

Can the word ke (that) be omitted here, just like we often do in English?