Magistra dicit discipulam nihil celare debere, si auxilium petere vult.

Because after dicit Latin is using an indirect statement construction, often called the accusative-and-infinitive.

In English we say:

• The teacher says that the student...

In Latin, instead of using that plus a finite verb, Latin often does this:

• magistra dicit discipulam ... debere

So:

• discipulam = the subject of the indirect statement, but put in the accusative
• debere = the verb of that indirect statement, put in the infinitive

Even though discipulam is logically the one who should hide nothing, grammatically it becomes accusative because it is inside indirect statement after dicit.

For the same reason: this is an indirect statement after dicit.

Latin does not normally say:

• Magistra dicit discipula nihil celare debet

to mean The teacher says that the student should hide nothing.

Instead it says:

• Magistra dicit discipulam nihil celare debere

So:

• debet = she ought
• debere = to ought / ought in indirect statement

A more natural English explanation is that Latin changes the verb into an infinitive after verbs of saying, thinking, knowing, hearing, and so on.

Debere means to owe, and from that it commonly comes to mean to ought, should, or must depending on context.

It is followed by another infinitive:

• celare = to hide
• debere = to ought / should

So celare debere means to ought to hide or more naturally should hide.

Inside the indirect statement:

• discipulam nihil celare debere = that the student should hide nothing

Nihil means nothing. Here it is the direct object of celare.

So:

• celare aliquid = to hide something
• celare nihil = to hide nothing

You can think of nihil as functioning like an accusative object, even though it is a special neuter form and does not always look like an ordinary noun.

So the structure is:

• discipulam = subject of the indirect statement
• nihil = object of celare
• celare debere = infinitive phrase meaning should hide

Because nihil celare is the natural and direct way to say to hide nothing.

Compare:

• nihil celare = to hide nothing
• non celare aliquid = literally not to hide something

The second version is much less natural if you mean nothing at all. Latin often uses a negative pronoun like nihil, nemo, or nullus rather than negating the verb and leaving the object indefinite.

So nihil celare is exactly what you would expect for hide nothing.

The subject of vult is understood to be discipula, the student.

The sentence means:

• The teacher says that the student should hide nothing, if she wants to ask for help.

So the person who wants is the same person as the one who should hide nothing: the student.

Latin often leaves subjects unstated when they are clear from context and from the verb ending.

• vult = he/she/it wants

Since discipulam is the relevant person in the sentence, we understand:

• si auxilium petere vult = if she wants to seek/ask for help

Because the si clause is not part of the indirect statement in the same way that discipulam nihil celare debere is.

The core indirect statement is:

• dicit discipulam nihil celare debere

Then Latin adds a condition:

• si auxilium petere vult = if she wants to ask for help

So vult stays finite because it is the verb of the conditional clause.

This is a useful point: in sentences like this, the accusative-and-infinitive construction can be followed by a subordinate clause with its own normal verb.

Because this is a simple condition: if she wants to ask for help.

Latin normally uses si with the indicative for an ordinary, real, or open condition. There is nothing especially hypothetical, contrary-to-fact, or indirect about the condition itself.

So:

• si ... vult = if she wants

That is exactly what you would expect.

Literally, petere means something like to seek, to aim at, to request, or to ask for, depending on context.

So:

• auxilium = help
• auxilium petere = to seek help, to ask for help

English often uses ask for help, but the Latin verb is broader than just ask. It can suggest actively seeking or requesting assistance.

Because petere takes a direct object.

So:

• petit auxilium = she asks for/seeks help

In the infinitive phrase:

• auxilium petere = to ask for help

That is why auxilium is accusative.

It does both.

• Magistra is the subject of dicit
• It also tells us the speaker is a female teacher

So the main clause is:

• Magistra dicit = The teacher says

Then what follows is what the teacher says:

• discipulam nihil celare debere

It can mean either, depending on context.

Latin present tense often covers both:

• says
• is saying

So magistra dicit could be translated as:

• The teacher says
• The teacher is saying

In a general statement, says is usually the most natural English choice.

Yes. Discipulam is the accusative singular of discipula, which means female student.

If the sentence meant a male student, you would expect:

• discipulum

So the Latin itself tells you the student is female.

That also helps us understand vult as she wants rather than he wants.