Post lectionem puella domi matri narrat se hodie didicisse cur quaedam praepositiones accusativum, aliae ablativum, quaedam utrumque accipiant, et cur idem verbum in indicativo aut in coniunctivo alium sensum habeat.

The main clause is puella domi matri narrat = the girl tells her mother at home.

Everything else hangs off that:

• post lectionem = a time phrase, after the lesson
• se hodie didicisse ... = an indirect statement, that she learned today ...
• inside that, there are two indirect questions introduced by cur:
  • cur quaedam praepositiones ... accipiant
  • et cur idem verbum ... habeat

So the sentence is built in layers:

1. The girl tells
2. that she learned
3. why X happens
4. and why Y happens

Because post is a preposition that takes the accusative.

So:

• lectio = nominative
• lectionem = accusative
• lectione = ablative

After post, Latin uses the accusative: post lectionem = after the lesson.

Domi is a special form meaning at home. It is the locative, a case/form used for place where, especially with certain words such as domus.

So:

• domi = at home
• domum = to home / homeward
• domo = from home

Latin often prefers domi rather than in domo when it simply means at home.

Because narrare commonly takes:

• a direct object = the thing told
• a dative = the person told

So matri narrat means she tells her mother.

That dative is the indirect object.

After verbs of saying, thinking, knowing, hearing, and similar verbs, Latin often uses the accusative-and-infinitive construction instead of a that-clause.

So:

• narrat = she says / tells
• se = the subject of the reported statement, in the accusative
• didicisse = the infinitive

Literally, it is something like:

she tells herself to have learned

But in natural English, that becomes:

she tells her mother that she learned

Because se is the reflexive pronoun. It refers back to the subject of the main verb, here puella.

So:

• puella ... narrat se didicisse = the girl tells ... that she herself learned
• eam would normally mean her, referring to some other female person, not the subject

This is a very common Latin pattern in indirect statement.

Because the learning happened before the telling.

• narrat = she tells now
• didicisse = to have learned, earlier than the telling

In indirect statement, the tense of the infinitive is relative to the main verb:

• present infinitive = same time
• perfect infinitive = earlier time
• future infinitive = later time

So se hodie didicisse means that she learned today or more literally that she has learned today.

Because they are in indirect questions.

The words cur ... accipiant and cur ... habeat do not give direct questions like:

• Cur quaedam praepositiones accusativum accipiunt?
• Cur idem verbum alium sensum habet?

Instead, they report what she learned about those questions. In Latin, indirect questions normally take the subjunctive.

So:

• cur ... accipiant
• cur ... habeat

are exactly what we expect.

A learner might expect past sequence because of didicisse, but here the sentence presents these points as general truths still valid now:

• some prepositions take the accusative
• others take the ablative
• the same word can have a different meaning in different moods

Because the content is still current and not just a one-time past action, the present subjunctive is natural.

In other contexts, especially if the writer wanted a more clearly past point of view, you could meet imperfect subjunctives in similar constructions. But here the present keeps the explanation lively and timeless.

It means:

• quaedam = some
• aliae = others
• quaedam = some

All three are feminine nominative plural, agreeing with praepositiones.

So the pattern is:

• quaedam praepositiones = some prepositions
• aliae = others
• quaedam = some

Latin often uses this kind of balanced pattern to divide a group into parts.

Because the names of cases are often used almost like nouns by themselves.

So:

• accusativum = the accusative
• ablativum = the ablative
• utrumque = both

The noun casum or casus is understood from the context.

So quaedam utrumque accipiant means some take both, that is, both cases.

Because they are doing different jobs.

• quaedam is the subject: some prepositions
• utrumque is the object: both = both cases

So there is no need for them to agree with each other.

You can think of it as:

some prepositions take both

where both is also singular in English, even though the subject is plural.

Latin could sometimes leave it out, but repeating it makes the structure clearer and more balanced.

The sentence is giving two separate things she learned:

• why some prepositions take different cases
• and why the same word has a different meaning in different moods

By repeating cur, Latin marks these as two parallel indirect questions.

Because Latin often omits a word when it is easy to understand from the context. This is called ellipsis.

The full idea is:

• quaedam praepositiones accusativum accipiant
• aliae ablativum accipiant
• quaedam utrumque accipiant

But after the first accipiant, the rest is obvious, so Latin leaves the repeated verb unspoken.

Here in means something like in the indicative or in the subjunctive, that is, when used in the indicative or when used in the subjunctive.

The nouns indicativo and coniunctivo are in the ablative because in with the ablative commonly expresses position or state.

So:

• in indicativo = in the indicative
• in coniunctivo = in the subjunctive

And aut simply links the two alternatives: or.

Because verbum is singular.

The subject of habeat is idem verbum = the same word, so the verb must also be singular:

• verbum ... habeat

Even though the previous part of the sentence had the plural subject praepositiones, this is a new clause with a new singular subject.

Latin often uses alius not only for another but also for different.

So alium sensum habere means:

• to have another meaning
• or more naturally, to have a different meaning

Thus idem verbum ... alium sensum habeat means that the same word can mean something different depending on whether it is in the indicative or the subjunctive.