Learn Hardcore German: Dass die Nachhilfe geholfen hat, gibt ihr neues Selbstvertrauen. - That the tutoring helped gives her new self‑confidence.

Dass die Nachhilfe geholfen hat, gibt ihr neues Selbstvertrauen.
→ The fact that the tutoring helped gives her new self-confidence.
→ It gives her new self-confidence that the tutoring helped. (more literal)

Why is the verb hat at the end of dass die Nachhilfe geholfen hat?

Because dass introduces a subordinate clause, and in German subordinate clauses the finite verb goes to the very end.

Main clause word order: Die Nachhilfe hat geholfen. (verb in 2nd position)
Subordinate clause with dass: …, dass die Nachhilfe geholfen hat. (finite verb at the end)

So:

geholfen is the past participle.
hat is the finite auxiliary verb, so it must come last in the subordinate clause.

What does Die Nachhilfe mean exactly?

Nachhilfe is a noun meaning tutoring, private lessons, or extra help (usually academic).

Nachhilfe in Mathe bekommen = to get extra tuition in math
Nachhilfelehrer = private tutor

So die Nachhilfe here is the tutoring / extra lessons she received.

What is the grammatical function of ihr here, and why is it ihr and not sie?

The verb geben (to give) usually has:

A dative indirect object: to whom?
An accusative direct object: what?

In this sentence:

gibt = gives
ihr = to her (dative)
neues Selbstvertrauen = new self-confidence (accusative)

That’s why it’s ihr (dative singular of sie = she), not sie:

sie (nominative) = she
ihr (dative) = to her

So literally:
Dass die Nachhilfe geholfen hat, gibt ihr neues Selbstvertrauen.
→ That the tutoring helped gives *to her new self-confidence.*

What case is neues Selbstvertrauen, and why does neues end in -es?

Neues Selbstvertrauen is the direct object of gibt, so it is in the accusative case.

Selbstvertrauen is:

a neuter noun: das Selbstvertrauen
here: accusative singular neuter

When a neuter noun in the accusative does not have an article (das, ein, etc.), the adjective takes a strong ending:

Nominative neuter: neues Selbstvertrauen
Accusative neuter: neues Selbstvertrauen

So neues here = new (accusative singular neuter, strong ending -es).

Is Selbstvertrauen always capitalized? And what does it mean?

Yes. Selbstvertrauen is a noun, so it is always capitalized in German.

Meaning:

Selbst = self
Vertrauen = trust

Together: self-confidence, self-assurance, literally self-trust.

Why is the main verb gibt (present) while the subordinate clause has geholfen hat (present perfect)?

The tenses reflect a time relation:

geholfen hat (present perfect) expresses a completed action in the past:
The tutoring has helped.
gibt (present) expresses a current, ongoing result:
…gives her new self-confidence (now).

So the idea is:

Because the tutoring has helped (completed), it now gives her new confidence (present result).

What exactly does Dass mean here, and how is it different from das?

Dass with double s is a subordinating conjunction meaning that (introducing a clause):

Ich weiß, dass die Nachhilfe geholfen hat.
= I know that the tutoring helped.

Das with one s can be:

the article das (the, neuter), or
the demonstrative pronoun das (that, this, it).

Example:

Das hat ihr geholfen. = That/This helped her.
Das Selbstvertrauen = the self-confidence

In your sentence, it must be dass (with ss) because it introduces the clause dass die Nachhilfe geholfen hat.

Why is there a comma after geholfen hat?

In German, subordinate clauses are separated from the main clause with a comma.

Dass die Nachhilfe geholfen hat, = subordinate clause (subject)
gibt ihr neues Selbstvertrauen. = main clause

Rule: when a sentence starts with a subordinate clause, you must place a comma before the main clause:

Wenn es regnet, bleibe ich zu Hause.
Weil ich müde bin, gehe ich früh ins Bett.
Dass die Nachhilfe geholfen hat, gibt ihr neues Selbstvertrauen.

Can I rewrite the sentence with normal word order, starting with Die Nachhilfe instead of the dass-clause?

Yes, you can express a very similar idea with a more straightforward subject:

Die Nachhilfe hat ihr neues Selbstvertrauen gegeben.
= The tutoring has given her new self-confidence.

This version:

makes Die Nachhilfe the subject of the sentence
uses the present perfect hat gegeben instead of gibt
avoids using a dass-clause as the subject

The original sentence, however, focuses more on the fact that the tutoring helped as the thing that gives her confidence.

Could I also say Weil die Nachhilfe geholfen hat, hat sie neues Selbstvertrauen? How is that different from the dass-version?

Yes, that is a grammatical but slightly different sentence:

Weil die Nachhilfe geholfen hat, hat sie neues Selbstvertrauen.
= Because the tutoring helped, she has new self-confidence.

Differences:

Weil gives a direct cause: because it helped, she has confidence.
Dass in the original sentence treats the fact that it helped as a thing that gives confidence:
- Dass die Nachhilfe geholfen hat, gibt ihr neues Selbstvertrauen.
  = The fact that the tutoring helped gives her new self-confidence.

So:

weil-sentence: focuses on reason
dass-sentence: focuses on the fact itself as a source of confidence

Is the word order … gibt ihr neues Selbstvertrauen fixed, or could I say … gibt neues Selbstvertrauen ihr?

The word order is not absolutely fixed, but there is a strong preference.

Most natural:

…, gibt ihr neues Selbstvertrauen. (dative ihr before accusative neues Selbstvertrauen)

You could say:

…, gibt neues Selbstvertrauen ihr.

but this sounds unusual or poetic and would normally require a special context or emphasis (for example, stressing ihr heavily).

Standard, neutral German normally places:

dative pronoun before the accusative object:
gibt ihr neues Selbstvertrauen.