Nuper avia dixit puellas saepius apud se manere, fratres autem rarius venire, quia procul habitant.

Because after dixit here Latin is using an indirect statement construction.

In English we say:

• Grandmother said that the girls stay...

In Latin, instead of that + finite verb, it often uses:

• accusative subject + infinitive

So:

• puellas ... manere = that the girls stay

That is why puellas is accusative rather than nominative.

For the same reason: they are part of an indirect statement after dixit.

Latin often does this after verbs of saying, thinking, knowing, hearing, and so on:

• dixit puellas manere
• literally: she said the girls to stay
• natural English: she said that the girls stay / were staying

So:

• puellas ... manere
• fratres ... venire

are two parallel indirect statements depending on dixit.

It actually can be accusative here — it just happens that in this noun the nominative plural and accusative plural have the same form.

For frater, fratris:

• nominative plural = fratres
• accusative plural = fratres

So in this sentence, fratres is understood as the accusative subject of venire, even though its form looks identical to the nominative.

Apud with the accusative often means at the house of, with, or in the company of someone.

So:

• apud se manere = to stay with her / at her place

Here it does not just mean physical nearness in a vague sense. It strongly suggests staying at grandmother’s home or with grandmother.

Because se is the reflexive pronoun.

It refers back to the subject of the main verb:

• avia dixit ... apud se manere
• Grandmother said ... to stay with her

The her here means the grandmother herself, so Latin uses se.

If Latin used eam, that would normally mean some other woman, not the grandmother.

This is a very common point for learners:

• se = self/same person as the subject
• eum/eam/eos/eas = someone else

Because they are comparative adverbs:

• saepe = often

• saepius = more often

• rare or more commonly raro = rarely
• rarius = more rarely / less often

The sentence is making a comparison between the two groups:

• the girls stay more often
• the brothers come less often

So the comparative forms fit naturally.

Autem is a contrast word, often translated as:

• however
• but
• on the other hand

So:

• puellas saepius ... manere, fratres autem rarius venire

means something like:

• the girls stay more often, but the brothers come less often

A useful thing to notice is that autem is usually not placed first in its clause. Latin often puts it in the second position or near the beginning after the word it contrasts with.

Nuper means recently or not long ago.

It tells you when the speaking happened:

• Nuper avia dixit = Recently grandmother said...

A learner may confuse it with something like newly, but in Latin nuper is a time adverb meaning a short time ago.

Because quia introduces a regular subordinate clause meaning because.

So after the indirect statements:

• puellas ... manere
• fratres ... venire

Latin then gives the reason in an ordinary clause:

• quia procul habitant = because they live far away

This is perfectly normal:

• dixit
  • indirect statement
• then quia
  • finite verb for the reason

Most naturally, it refers to fratres, the nearest plural group:

• the brothers come less often, because they live far away

That is probably the intended sense.

Could it theoretically refer to both groups? In isolation, a plural verb can sometimes leave a little ambiguity. But in this sentence the most straightforward reading is:

• girls stay more often with grandmother
• brothers come less often
• because the brothers live far away

So a learner should usually take habitant as referring to fratres.

Because Latin word order is much more flexible than English word order. The endings carry much of the grammar, so Latin can arrange words for emphasis and clarity.

Here the sentence is structured to show the contrast clearly:

• Nuper avia dixit
• puellas saepius apud se manere
• fratres autem rarius venire
• quia procul habitant

This lets Latin place side by side:

• puellas ... saepius
• fratres autem ... rarius

So the sentence highlights the contrast between the girls and the brothers.

Yes. In indirect statement, the present infinitive usually shows action that is simultaneous with the main verb.

So after dixit:

• puellas manere
• fratres venire

means that, at the time grandmother spoke, she said these things were true:

• the girls stay / are staying
• the brothers come / are coming

In natural English, depending on context, you might translate more freely, but the Latin present infinitive is the normal way to express contemporaneous action in indirect statement.