Learn Hardcore Swedish: Hon köper nya skor eftersom de gamla är trasiga. - She buys new shoes because the old ones are broken.

Because adjectives agree with the noun in number (and gender/definiteness). In the plural indefinite, adjectives take the ending -a. So with plural skor (shoes), you must use nya.

en ny sko (a new shoe)
ett nytt hus (a new house)
nya skor / nya hus (new shoes / new houses)

What does "de gamla" mean without a noun?

It’s an ellipsis meaning the old ones. It stands for de gamla skorna (the old shoes), but since the noun is obvious from context, Swedish can drop it and keep de + adjective. This is related to Swedish “double definiteness”: with a definite noun and an adjective, you use both a separate article and the noun’s definite ending (de gamla skorna). When the noun is omitted, you still write de gamla.

Why is it "de gamla" and not "dem gamla"?

When you have de + adjective (+ [omitted] noun), you always write de, regardless of whether the whole phrase is subject or object:

Jag slängde de gamla. (I threw the old ones away.) — correct
Jag slängde dem. (I threw them away.) — also correct, but here there’s no adjective.

Using dem in dem gamla is a very common mistake in writing influenced by the spoken form “dom.”

How do you pronounce "de" here?

In almost all modern Swedish, both de and dem are pronounced dom. In informal writing you may see dom, but in standard writing you should use de/dem according to the grammar rules above.

Why is it "trasiga" and not "trasig"?

Predicate adjectives agree in number with the subject. The understood subject is plural (the shoes), so you use plural trasiga:

Skorna är trasiga. (The shoes are broken.)
Skon är trasig. (The shoe is broken.)
Huset är trasigt. (The house is broken/damaged.)

What’s the difference between "eftersom," "för att," and "därför att"?

eftersom = because/since (reason). Unambiguous and fine both clause-initial and after the main clause.
för att can mean either “in order to” (purpose) or “because” (reason). As “because,” it usually comes after the main clause and can be ambiguous.
därför att = explicitly “because.” Often used to remove the ambiguity of för att.
eftersom att is heard in some dialects/colloquial speech but is avoided in formal writing.

Examples:

Hon köper nya skor, för att de gamla är trasiga. (because)
Hon köper nya skor för att spara tid. (in order to)
Hon köper nya skor, därför att de gamla är trasiga. (clearly “because”)
Eftersom de gamla är trasiga köper hon nya skor. (because)

Why is it "de gamla är" and not "är de gamla" after "eftersom"?

Swedish subordinate clauses (introduced by eftersom, att, när, etc.) do not use main-clause inversion. The typical order is:

Subordinator + Subject + Finite verb + ... Hence: eftersom de gamla är trasiga (not “eftersom är de gamla trasiga”).

Can I start the sentence with the reason clause?

Yes: Eftersom de gamla är trasiga, köper hon nya skor. When a subordinate clause comes first, the following main clause uses V2 word order, so the finite verb comes before the subject: köper hon (not “hon köper” in that position).

Do I need a comma before "eftersom"?

No comma is needed when the eftersom-clause follows the main clause: Hon köper nya skor eftersom … If the eftersom-clause comes first, put a comma after it: Eftersom de gamla är trasiga, köper hon nya skor.

What nuance does the present tense "köper" have? Does it mean “is buying” or “buys”?

Swedish present tense covers both English simple and progressive meanings. Hon köper can mean “she is buying” (right now) or “she buys” (habitually), depending on context. If you want to stress intention/future, you can say Hon ska köpa nya skor (“she is going to buy”).

Why are the new shoes indefinite but the old ones definite?

The new shoes are not specific/known yet, so you use the indefinite plural nya skor. The old ones are specific (the ones she already owns), so you use the definite phrase de gamla (“the old ones” = de gamla skorna).

What are the forms of the noun "sko" (shoe)?

Singular indefinite: en sko
Singular definite: skon
Plural indefinite: skor
Plural definite: skorna

How do you pronounce "köper" and "skor"?

köper: the k before a front vowel (ö) is the “soft k,” pronounced roughly like English “sh/ch” in Swedish: approximately [ˈɕøːpɛr]. The vowel ö is like French “eu” in “deux.”
skor: [skuːr]. Here sk is a regular [sk] (not the “sj” sound) because it’s before o (a back vowel), and the vowel is a long [uː].

Could I use "sönder" instead of "trasiga"?

In everyday Swedish, yes: De gamla är sönder or Skorna är sönder is common. More standard is trasig/trasiga. To express the breaking event, use the verb phrase gå sönder: Skorna har gått sönder (The shoes have broken).

Is "de gamla" ever used about people?

Yes. De gamla can mean “the elderly” when the context is people. In this sentence, the context is shoes, so de gamla = “the old ones (shoes).” Context determines the reference.