Die Jacke hängt schon auf dem Kleiderbügel, aber ihr Reißverschluss klemmt wieder.

In German, hängen is the normal verb for something that is in a hanging position.

• Die Jacke hängt am Haken. = The jacket is hanging on the hook.
• Das Bild hängt an der Wand. = The picture is hanging on the wall.

English often uses is plus hanging, but German usually just uses the verb hängen by itself.

Because auf is a two-way preposition:

• it takes the accusative when there is movement toward a destination
• it takes the dative when it describes a location

Here, the jacket is already there, so this is a location, not movement:

• Die Jacke hängt auf dem Kleiderbügel. = The jacket is hanging on the hanger.

If you were talking about putting it there, you would use accusative:

• Ich hänge die Jacke auf den Kleiderbügel. = I hang the jacket onto the hanger.

Kleiderbügel is a masculine noun: der Kleiderbügel.

After auf in a location meaning, you need the dative. The dative form of der is dem.

So:

• nominative: der Kleiderbügel
• dative: dem Kleiderbügel

That is why the sentence has auf dem Kleiderbügel.

Here schon means already.

So Die Jacke hängt schon auf dem Kleiderbügel suggests that the jacket is already on the hanger, perhaps implying that one step has been done, but there is still a problem.

Depending on tone and context, schon can have other meanings in German, but in this sentence already is the natural one.

Here ihr means its, referring back to die Jacke.

This is something English speakers often notice, because German possessive words agree with the grammatical gender of the owner, not with biological sex.

• die Jacke is feminine
• so her/its zipper becomes ihr Reißverschluss

Other examples:

• das Auto und sein Motor = the car and its engine
• der Tisch und sein Bein = the table and its leg
• die Jacke und ihr Reißverschluss = the jacket and its zipper

So ihr here does not mean a female person owns the zipper. It just matches the feminine noun Jacke.

Because aber is a coordinating conjunction. It connects two main clauses, and each clause keeps normal main-clause word order.

So:

• Die Jacke hängt schon auf dem Kleiderbügel
• aber ihr Reißverschluss klemmt wieder

In the second clause, the finite verb klemmt is still in second position.

Compare that with a word like deshalb, which does count as the first element and causes inversion:

• Die Jacke hängt schon auf dem Kleiderbügel. Deshalb klemmt ihr Reißverschluss wieder.

But with aber, no inversion happens.

Klemmen means something like:

• to be stuck
• to jam
• to catch
• to not move properly

For a zipper, Der Reißverschluss klemmt means the zipper is not working smoothly and gets stuck.

It is a very natural verb for mechanical things that do not move the way they should.

Examples:

• Die Tür klemmt. = The door sticks.
• Die Schublade klemmt. = The drawer is jammed.
• Der Reißverschluss klemmt. = The zipper is stuck.

Wieder here means again.

So klemmt wieder suggests that this is a repeated problem:

• the zipper is stuck again
• it has happened before

German usually places wieder quite naturally near the verb or later in the clause, as here.

Yes, Reißverschluss is the normal everyday word for zipper.

It is a compound noun:

• reißen = to tear / pull apart
• Verschluss = closure / fastening

You do not need to analyze it every time, but it is useful to know that German often builds long nouns by combining smaller words.

Also remember that it is masculine:

• der Reißverschluss

So:

• der Reißverschluss klemmt
• ein kaputter Reißverschluss
• den Reißverschluss öffnen

German main clauses usually put the verb in second position, but the first position can be filled by different elements.

Here the subject comes first:

• Die Jacke hängt schon auf dem Kleiderbügel

That is the most neutral order.

You could move another element to the front for emphasis, but then the verb would still stay second:

• Schon hängt die Jacke auf dem Kleiderbügel.
• Auf dem Kleiderbügel hängt die Jacke schon.

These are possible, but less neutral in everyday use.

Yes. German learners often meet two closely related uses of hängen:

1. intransitive: something is hanging

   • Die Jacke hängt am Haken.

2. transitive: someone hangs something up

   • Ich hänge die Jacke an den Haken.

In the present tense, the forms often look the same, so the difference is mainly in the meaning and in whether there is a direct object.

In your sentence, Die Jacke hängt is the intransitive use: the jacket is in a hanging position.

Sometimes learners wonder about this because English often says on a hanger.

In German, auf dem Kleiderbügel is perfectly natural when the jacket is on the hanger.
You may also hear related expressions depending on how the speaker imagines the position, but auf dem Kleiderbügel is a very standard way to say it.

So for learning purposes, this phrase is good and idiomatic as it stands.

Because once die Jacke has already been mentioned, German can refer back to it with a possessive phrase:

• ihr Reißverschluss = its zipper

This is efficient and natural. Instead of saying something like aber der Reißverschluss der Jacke, German uses the shorter possessive structure.

Both are possible in principle:

• aber ihr Reißverschluss klemmt wieder
• aber der Reißverschluss der Jacke klemmt wieder

The version with ihr sounds smoother and more natural here.