Después de secarse, la camisa quedó tan arrugada que tuve que pasarle la plancha.

Because secarse here means to dry / to get dry as an intransitive, pronominal verb. The subject is la camisa.

So:

• después de secarse = after drying / after it dried
• después de secarla = after drying it (someone dried the shirt)

In this sentence, the idea is that the shirt went through the drying process and then ended up wrinkled. Spanish often uses secarse for clothes, hair, paint, etc. when they dry.

Example:

• La ropa se secó al sol. = The clothes dried in the sun.
• Sequé la ropa. = I dried the clothes.

Not really in the literal English reflexive sense.

Even though se often looks reflexive, with many verbs it forms a common pronominal verb that English does not translate reflexively. So here la camisa se secó is just a natural way to say the shirt dried or the shirt got dry.

A learner might think:

• secar = to dry something
• secarse = to dry oneself

That is true in some contexts, but with objects like clothes, secarse is very commonly just to dry.

Quedar + adjective is a very common way to express a resulting state in Spanish.

So la camisa quedó arrugada means something like:

• the shirt ended up wrinkled
• the shirt was left wrinkled

It does more than simply describe the shirt; it tells you the condition was the result of what happened before.

That makes it a very good fit after después de secarse.

The difference is mainly result vs description.

• quedó arrugada = it ended up wrinkled, or became wrinkled as a result
• estaba arrugada = it was wrinkled; this just describes the condition

So in this sentence, quedó is better because the drying process caused the wrinkled state.

Compare:

• Después de secarse, la camisa quedó arrugada.
  The shirt ended up wrinkled after drying.
• La camisa estaba arrugada.
  The shirt was wrinkled.

This is the Spanish pattern:

tan + adjective + que

It means so + adjective + that.

So:

• tan arrugada que tuve que... = so wrinkled that I had to...

More examples:

• Estaba tan cansado que me dormí. = I was so tired that I fell asleep.
• Hacía tanto calor que abrimos las ventanas. = It was so hot that we opened the windows.

Note that with adjectives you use tan:

• tan arrugada
• tan difícil
• tan rápido

Because tuve que presents the obligation as a completed, specific event in the past.

Here, the idea is:

• I saw the shirt was very wrinkled
• so I had to iron it

That is a one-time necessity, so the preterite tuve is natural.

Compare:

• Tuve que pasarle la plancha.
  I had to iron it.
  A specific thing I had to do on that occasion.

• Tenía que pasarle la plancha todos los días.
  I had to iron it every day.
  Habitual or ongoing obligation.

Both are possible.

• plancharla = to iron it
• pasarle la plancha = literally to pass the iron over it

In Spain, pasar la plancha or pasarle la plancha a algo is very natural and idiomatic. It sounds a little more descriptive, while plancharla is simpler and more direct.

So these are both fine:

• Tuve que plancharla.
• Tuve que pasarle la plancha.

The version in your sentence is especially common in everyday speech.

Le refers to la camisa.

Structure:

• pasar algo a alguien/algo
• here: pasar la plancha a la camisa
• so a la camisa becomes le

That gives:

• pasarle la plancha = pass the iron over the shirt

This can be confusing because la camisa is feminine, but the pronoun is le, not la, because it is functioning like an indirect object here.

So:

• la would be the direct object in plancharla
• le is the indirect object in pasarle la plancha

Because Spanish normally uses the definite article with many everyday objects and instruments when they are understood in context.

So la plancha here means the iron.

Spanish often says:

• lavarse las manos
• cerrar la puerta
• pasar la aspiradora
• pasar la plancha

Even when English might omit an article or use a possessive, Spanish often prefers the definite article.

It is historically the past participle of arrugar (to wrinkle), but in this sentence it is functioning as an adjective.

That is why it agrees with la camisa:

• feminine singular noun: la camisa
• feminine singular adjective: arrugada

Other forms:

• el pantalón arrugado
• las camisas arrugadas
• los pantalones arrugados

So yes, it comes from a participle, but here you should think of it as an adjective meaning wrinkled.

Yes.

You could say something like:

• Después de que se secara, la camisa quedó tan arrugada que tuve que pasarle la plancha.

But después de + infinitive is shorter and very common when the subject is clear from context.

So:

• después de secarse = after drying
• después de que se secara = after it dried

Both are grammatical, but the infinitive version is more compact and very natural here.

It could be understood, but quedó tan arrugada is the more natural choice here.

• quedó arrugada focuses on the result: it ended up wrinkled
• se quedó arrugada can sound more like it remained wrinkled or it stayed wrinkled

With objects like clothes, quedar + adjective is usually the cleaner, more idiomatic option for this kind of result.

Yes, especially because of pasarle la plancha, which sounds very natural in Spain.

A speaker from another region might be more likely to say:

• tuve que plancharla

But the full sentence is perfectly natural and standard:

• Después de secarse, la camisa quedó tan arrugada que tuve que pasarle la plancha.

It sounds like normal everyday Spanish from Spain.