Learn Hardcore German: Die, die im Garten sitzen, hören Musik. - Those who sit in the garden listen to music.

The first Die (at the very start) stands alone before a comma. That is the demonstrative pronoun: “Those” / “The ones”.
The second die starts the part between the commas: die im Garten sitzen. That is a relative clause, and this die is the relative pronoun: “who”.

Anything between the commas here is extra information describing Die. That is typical of a relative clause in German.

What exactly is a relative clause in this sentence?

The relative clause is:

die im Garten sitzen

It describes which ones we are talking about: those who are sitting in the garden.

It starts with a relative pronoun (die) = who.
It has its own verb at the end (sitzen).
It is set off by commas from the main clause.

Main clause: Die … hören Musik.
Relative clause: die im Garten sitzen (describing Die).

Why is there a comma before and after die im Garten sitzen?

In German, all relative clauses must be separated by commas from the rest of the sentence.

Comma before: marks the start of the relative clause.
Comma after: marks the end of the relative clause.

So in Die, die im Garten sitzen, hören Musik., the commas show that die im Garten sitzen is a relative clause giving extra information about Die.

What is the grammatical function of Die in the main clause?

The first Die is the subject of the main clause:

Die (subject)
… hören (finite verb)
Musik (direct object)

Even though there is a relative clause in between, the core structure is still:

Die hören Musik. – Those (people) are listening to music.

Is die here feminine singular or plural? How do I know?

In this sentence, die is nominative plural:

It refers to several people: Those (people) who are in the garden…
The verb hören is in the 3rd person plural form, matching a plural subject.

So both Die (demonstrative) and die (relative pronoun) are nominative plural.
If it were feminine singular, the verb would be hört:
Die, die im Garten sitzt, hört Musik. – The woman who sits in the garden is listening to music.

Could I say this sentence with a clear noun instead of just Die?

Yes. A fully explicit version would be:

Die Leute, die im Garten sitzen, hören Musik.
The people who are sitting in the garden are listening to music.

Here:

Die Leute = “the people”
die im Garten sitzen = “who are sitting in the garden”

Your original sentence just leaves Leute implicit and uses Die = “Those (people)” / “The ones”.

Can I drop the first Die and just say Die im Garten sitzen, hören Musik?

You will sometimes see:

Die im Garten sitzen, hören Musik.

This is also possible and understandable. Grammatically, the Die im Garten sitzen together functions as the subject, roughly like “Those who are sitting in the garden”.

However:

With Die, die im Garten sitzen, …, the demonstrative Die is emphasized a bit more: “Those (people), who are sitting in the garden, …”
Without the first Die, it feels a little more compressed and less emphatic.

Both forms are acceptable in modern German.

Why is the verb sitzen at the end of die im Garten sitzen?

Because die im Garten sitzen is a subordinate clause (a relative clause).
In German subordinate clauses, the conjugated verb goes to the end.

Pattern:

Main clause: Sie sitzen im Garten. – They sit in the garden. (verb is 2nd)
Relative clause: … die im Garten sitzen. – … who sit in the garden. (verb is last)

So the word order im Garten sitzen is required by the grammar of subordinate clauses.

Can I use welche instead of the second die, like Die, welche im Garten sitzen, …?

You can, but it sounds more formal or old-fashioned in everyday German:

Die, welche im Garten sitzen, hören Musik.

Here welche is also a relative pronoun (like die = who), but in modern spoken German, people almost always use der / die / das as relative pronouns.

So the most natural version is still:

Die, die im Garten sitzen, hören Musik.

How would the sentence change in the singular?

For a single person, you adjust the pronouns and the verb:

Masculine:
- Der, der im Garten sitzt, hört Musik.
  The man who is sitting in the garden is listening to music. / He who sits in the garden listens to music.
Feminine:
- Die, die im Garten sitzt, hört Musik.
  The woman who is sitting in the garden is listening to music.
Neuter:
- Das, das im Garten sitzt, hört Musik. (unusual for a person, more for things/animals with das)

Note:

First word: Der / Die / Das = demonstrative (“the one who”).
Second: der / die / das = relative pronoun (“who / that”).
Verb in singular: sitzt, hört.

Why is Die capitalized the first time but not the second time?

Because:

The first Die is at the beginning of the sentence, and every sentence in German starts with a capital letter.
The second die is inside the sentence, so it is written with a lowercase d (unless it happens to start a new sentence, which it doesn’t here).

They are the same word form (pronoun die), just capitalized because of normal sentence rules, not because of a special grammatical difference.