Learn Farsi: این دختران خوانندگان جدید هستند.

The singular word for girl is dokhtar. Because a girl is a living (animate) being, we use the animate plural marker -an. You just attach it to the end: dokhtar + -an = dokhtaran.

Why does the word for singers (khanandegan) use -gan instead of just -an?

The singular word is khanandeh (singer), which ends in a silent h (pronounced as a short e). When adding the animate plural ending -an to words ending in this vowel sound, a g is inserted to make it easier to say. So, khanandeh becomes khanandegan.

The translation says These girls, so why does the sentence start with the singular word in (this)?

In Farsi, pointing words like in (this) do not become plural when they are placed right next to a noun. The plural noun dokhtaran already shows we mean more than one, so Farsi just says in dokhtaran (literally this girls).

Since the singers are plural, shouldn't the adjective jadid (new) be plural too?

No, in Farsi, adjectives generally stay in their basic, singular form even when they describe a plural noun. It stays as jadid, not jadidan.