Jag brukar hänga min jacka på en galge så att den inte blir skrynklig.

Brukar is used to talk about something that someone usually, typically, or habitually does.

So Jag brukar hänga min jacka... means I usually hang my jacket... or I tend to hang my jacket...

A useful pattern is:

• jag brukar + infinitive

Examples:

• Jag brukar läsa på kvällen. = I usually read in the evening.
• Hon brukar cykla till jobbet. = She usually bikes to work.

Because after brukar, Swedish uses the infinitive form of the next verb.

So:

• brukar hänga = usually hang
• not brukar hänger

This is similar to English:

• I usually hang
• not I usually hangs

Compare:

• Jag hänger min jacka där. = I hang my jacket there.
• Jag brukar hänga min jacka där. = I usually hang my jacket there.

In Swedish, when you use a possessive word like min, din, hans, hennes, vår, the noun usually does not take the definite ending.

So:

• min jacka = my jacket
• not min jackan

Compare:

• jacka = jacket
• jackan = the jacket
• min jacka = my jacket

More examples:

• min bok = my book
• hennes bil = her car
• vårt hus = our house

En galge means a hanger or a clothes hanger.

So:

• på en galge = on a hanger

It is a common-gender noun, so it takes en:

• en galge
• galgen = the hanger
• galgar = hangers

Swedish uses på with galge in this expression, so hänga något på en galge means to hang something on a hanger.

This is just the normal preposition used here:

• hänga jackan på en galge

Similar examples:

• hänga tavlan på väggen = hang the painting on the wall
• hänga skjortan på en galge = hang the shirt on a hanger

Prepositions do not always match English perfectly, so it is best to learn the whole phrase together.

Så att means so that.

It introduces a clause showing purpose or result.

So:

• Jag brukar hänga min jacka på en galge så att den inte blir skrynklig.
• I usually hang my jacket on a hanger so that it doesn’t get wrinkled.

More examples:

• Jag skriver upp det så att jag inte glömmer. = I write it down so that I don’t forget.
• Hon talade långsamt så att alla förstod. = She spoke slowly so that everyone understood.

Den refers to jacka.

Since jacka is an en-word, the pronoun used for it is den.

• en jacka → den
• ett hus → det

So in this sentence:

• den inte blir skrynklig = it doesn’t get wrinkled
• den = the jacket

Example:

• Jag köpte en bok. Den är bra. = I bought a book. It is good.
• Jag köpte ett bord. Det är stort. = I bought a table. It is big.

Because så att den inte blir skrynklig is a subordinate clause.

In Swedish subordinate clauses, sentence adverbs like inte usually come before the finite verb.

So:

• main clause: Den blir inte skrynklig.
• subordinate clause: ...så att den inte blir skrynklig.

This is a very important word-order rule in Swedish.

Compare:

• Jag vet att han inte kommer. = I know that he is not coming.
• Han kommer inte. = He is not coming.

Because bli means become or get, while vara means be.

Here the idea is not that the jacket is wrinkly already, but that it doesn’t become/get wrinkly.

So:

• inte blir skrynklig = doesn’t get wrinkled
• inte är skrynklig = isn’t wrinkled

The second one describes a state. The first one describes a change into that state.

Skrynklig means wrinkled, creased, or crumpled-looking, especially about clothes or fabric.

So:

• en skrynklig skjorta = a wrinkled shirt
• jackan blir skrynklig = the jacket gets wrinkled

Since jacka is an en-word, the common singular form is:

• skrynklig

Compare adjective agreement:

• en skrynklig jacka
• ett skrynkligt papper
• skrynkliga kläder

Yes, hänga is a common verb meaning to hang.

Basic forms:

• infinitive: hänga
• present: hänger
• past: hängde
• supine: hängt

Examples:

• Jag hänger jackan här. = I hang the jacket here.
• Jag hängde jackan i hallen. = I hung the jacket in the hallway.
• Jag har hängt jackan där. = I have hung the jacket there.

Sometimes, but not in exactly the same way.

In this sentence, så att works naturally because it is followed by a full clause:

• så att den inte blir skrynklig

För att is often used more like in order to. If you use it, the structure may change. For example:

• Jag brukar hänga min jacka på en galge för att den inte ska bli skrynklig.

That is also possible, but it sounds a bit more explicitly purposeful.

A simple learner rule is:

• så att + clause = so that
• för att + infinitive/clause = in order to / for

The sentence has two parts:

1. Jag brukar hänga min jacka på en galge

   • Jag = subject
   • brukar = finite verb
   • hänga = infinitive
   • min jacka = object
   • på en galge = prepositional phrase

2. så att den inte blir skrynklig

   • subordinate clause introduced by så att
   • den = subject
   • inte = sentence adverb
   • blir = finite verb
   • skrynklig = adjective

So the whole sentence is:

• main clause + purpose/result clause

It sounds natural and idiomatic.

It expresses a normal everyday habit and a practical reason for it:

• Jag brukar... = habitual action
• så att... = reason/purpose/result

A native speaker might also say similar things like:

• Jag hänger alltid min jacka på en galge så att den inte blir skrynklig.
• Jag brukar hänga upp min jacka på en galge så att den håller sig fin.

But your sentence is fully natural Swedish.