Discipula respondet se tandem intellegere et libenter discere.

Because discipula is the subject of the main verb respondet. In Latin, the subject of a finite verb is normally in the nominative case.

So here:

• discipula = the female student / schoolgirl
• respondet = answers / replies

That gives the basic main clause: The student replies.

Respondet is:

• 3rd person singular
• present tense
• active voice
• indicative mood

It comes from respondere, meaning to answer or to reply.

So discipula respondet means the student replies or the student is replying, depending on context.

Se is the reflexive pronoun. It refers back to the subject of the main clause, which is discipula.

So the sentence means that the student says something about herself:

• se intellegere = that she understands
• not that some other woman understands

A learner might expect a form that clearly looks feminine, but se does not change for gender. It can mean himself, herself, or themselves, depending on who it refers to.

Because this is an example of the Latin indirect statement construction, often called accusative + infinitive.

After verbs of saying, thinking, knowing, perceiving, and similar ideas, Latin often uses:

• an accusative subject
• plus an infinitive

So in se tandem intellegere et libenter discere:

• se is the subject of the infinitives
• intellegere and discere are the infinitives

Even though se is logically the subject of understand and learn, in Latin it appears in the accusative because that is how indirect statement works.

Yes. Se tandem intellegere et libenter discere is an indirect statement after respondet.

English often uses that:

• The student replies that she finally understands and gladly learns
• or more naturally, The student replies that she finally understands and is happy to learn

Latin often does not use a separate word for that here. Instead, it uses the accusative-and-infinitive construction:

• se = subject of the indirect statement
• intellegere / discere = infinitives

They are infinitives because they are part of the indirect statement.

If Latin used ordinary finite verbs, it would be a different structure. But after respondet, Latin commonly switches into indirect statement, where the reported action is expressed with infinitives.

So:

• respondet se intellegere = she replies that she understands
• not a structure like respondet quod intellegit, which is possible in some kinds of Latin but is not the standard classical pattern here

Yes. Se is understood as the subject of both infinitives.

So the sentence means:

• she finally understands
• and she gladly learns

Latin does not need to repeat se before the second infinitive, because the same subject continues across both verbs.

Tandem usually means at last, finally, or after all this time.

In this sentence it most naturally goes with intellegere:

• se tandem intellegere = that she finally understands

It gives a sense that understanding has taken some time or effort.

Libenter means gladly, willingly, or with pleasure.

It is an adverb because it modifies the verb discere:

• libenter discere = to learn gladly / willingly

A native English speaker might first think of an adjective like happy, but Latin often expresses this idea with an adverb:

• I gladly learn rather than
• I am happy learning

Basically, discere means to learn. In some contexts, it can sound a bit like to study, especially in school settings, but its core meaning is to learn.

So libenter discere can suggest:

• to learn willingly
• to enjoy learning
• to study gladly

The exact English wording depends on context.

Because in this kind of sentence Latin usually does not need one.

English says:

• She replies that she understands

Latin often says this by using indirect statement:

• respondet se intellegere

So the idea of that is built into the structure itself, rather than being expressed by a separate word.

Yes. Verbs of replying, saying, reporting, and similar communication can introduce indirect statement in Latin.

So respondet se tandem intellegere et libenter discere is a normal way to say that the student replies with the content:

• that she finally understands
• and gladly learns

A very literal unpacking would be:

• Discipula = the female student
• respondet = replies
• se = herself
• tandem = finally
• intellegere = to understand
• et = and
• libenter = gladly
• discere = to learn

So, very literally:

The student replies herself finally to understand and gladly to learn.

That sounds bad in English, but it helps show the Latin structure. Natural English would be something like:

The student replies that she finally understands and is happy to learn.

Yes. Both spellings are found.

You may see:

• intellegere
• intelligere

The form with e, intellegere, is often preferred in modern editions for classical Latin, but intelligere is also very common in textbooks and dictionaries. They are the same verb.

So in this sentence, intellegere simply means to understand.