Learn Arabic (Modern Standard): كانت فاطمة قد رأت فراشة في غرفتها.

Using the past tense of كان (to be) followed by another past tense verb creates the past perfect tense, meaning that someone 'had done' something. Here, كانت and رأت combine to mean 'had seen'.

What is the purpose of the word قد here?

When placed before a past tense verb, قد emphasizes that the action was fully completed. In this past perfect structure, combining كان with قد and a past tense verb is the standard Arabic way to express that an action had already happened.

Why does the subject فاطمة separate the two verbs rather than coming before or after them together?

In Arabic compound tenses, the subject usually comes immediately after the first verb. Because كانت is the first verb acting as the helper, فاطمة follows it directly before the rest of the verb phrase قد رأت.

Why does the verb رأت (she saw) drop the final letter found in its root رأى (he saw)?

This happens with verbs ending in a weak letter like alif maqsurah. When you add the feminine suffix ـت (which carries no vowel), the weak letter drops out completely so you do not have two unvoweled consonants right next to each other.