Pater dicit fenestram fractam statim reficiendam esse, ne stillae pluviae intus cadant.

Because after dicit, Latin is using indirect statement. In this construction, the “subject” of the reported statement goes into the accusative, and the verb goes into the infinitive.

So:

• direct statement: fenestra fracta statim reficienda est
  • the broken window must be repaired at once
• indirect statement after dicit: pater dicit fenestram fractam statim reficiendam esse
  • father says that the broken window must be repaired at once

So fenestram is accusative because it is the subject of the infinitive phrase.

It is the very common Latin construction called indirect statement (also called accusative and infinitive).

The pattern is:

• a main verb of saying, thinking, knowing, perceiving, etc.
• then an accusative
• then an infinitive

Here:

• Pater dicit = Father says
• fenestram ... reficiendam esse = that the window must be repaired

So literally, Latin says something like:

• Father says the broken window to-be-repaired

which English turns into:

• Father says that the broken window must be repaired

Because fractam agrees with fenestram.

Fenestram is:

• feminine
• singular
• accusative

So any adjective describing it must match those same features. Fractam is the feminine singular accusative form of fractus, fracta, fractum, meaning broken.

So:

• fenestram fractam = the broken window

Fractam is the perfect passive participle of frangere, meaning to break.

So fractam means:

• broken
• more literally, having been broken

In normal English, the broken window is the natural translation.

Reficiendam esse is a gerundive + esse construction, often called the passive periphrastic. It expresses necessity, duty, or obligation.

So:

• reficere = to repair
• reficiendam esse = to have to be repaired, must be repaired

This is not just the idea of repairing in general; it specifically means that the window needs repairing.

Because the sentence is in indirect statement after dicit, Latin uses:

• reficiendam esse instead of
• reficienda est

Compare:

• direct: fenestra reficienda est = the window must be repaired
• indirect: dicit fenestram reficiendam esse = he says that the window must be repaired

Because it also agrees with fenestram.

The gerundive here functions like an adjective and must match the noun it goes with:

• fenestram = feminine singular accusative
• reficiendam = feminine singular accusative

So both fractam and reficiendam describe fenestram:

• fenestram fractam statim reficiendam esse
• that the broken window must be repaired at once

Mostly because Latin word order is flexible, and it often places the infinitive esse late in the clause, especially in indirect statement.

Latin does not depend on word order as strictly as English does, because the endings already show the grammar.

So this is normal Latin style:

• fenestram fractam statim reficiendam esse

You could rearrange parts somewhat without changing the basic meaning, though some orders sound more natural than others.

Statim means at once, immediately. It modifies the idea of reficiendam esse:

• must be repaired immediately

So the sense is not just that the window must be repaired, but that it must be repaired right away.

This is a negative purpose clause.

• ne = so that ... not, lest
• cadant = subjunctive verb
• the whole clause gives the purpose of repairing the window

So:

• fenestram ... reficiendam esse, ne stillae pluviae intus cadant
• the window must be repaired, so that raindrops do not fall inside
• or more naturally: ... lest raindrops fall inside

In Latin, a negative purpose clause is normally introduced by ne, not ut non.

So:

• ut = positive purpose: so that
• ne = negative purpose: so that ... not, lest

Thus:

• ne stillae pluviae intus cadant = lest raindrops fall inside

Because it is in a purpose clause introduced by ne.

Purpose clauses regularly use the subjunctive in Latin. So:

• ne ... cadant = lest ... fall

The present subjunctive here fits the idea of an action that is intended to be prevented now or in the immediate future.

Stillae is the subject of cadant:

• stillae = drops

Pluviae is a genitive depending on stillae:

• pluviae = of rain

So together:

• stillae pluviae = drops of rain
• more naturally in English: raindrops

Intus means inside or indoors. It is an adverb.

That means it does not take an object here; it simply tells where the raindrops would fall:

• intus cadant = fall inside

The core reported statement would become:

• fenestra fracta statim reficienda est, ne stillae pluviae intus cadant

That means:

• The broken window must be repaired at once, lest raindrops fall inside

Then the original sentence simply puts that idea after pater dicit and changes the grammar into indirect statement:

• fenestra → fenestram
• reficienda est → reficiendam esse

The main asserted idea is in reficiendam esse:

• the window must be repaired

Fractam is descriptive information about the window:

• the broken window

So the structure is roughly:

• fenestram = the noun
• fractam = descriptive adjective/participle
• reficiendam esse = the essential predication, what is being said about the window

In other words, Father is not mainly saying the window is broken; he is saying the broken window needs to be repaired immediately.