Después de la playa, tiendo la toalla en el balcón para que se seque.

Because después normally needs de before a noun.

So you say:

• después de la playa
• después de comer
• después de la cena

Without de, it sounds incomplete before a noun.

You can use después by itself only when nothing follows it:

• Voy ahora; tú vienes después.

Yes. It is a very natural, shortened way to mean after going to the beach or after being at the beach.

Spanish often leaves out parts that are understood from context. So después de la playa can imply:

• after the beach trip
• after coming back from the beach
• after spending time at the beach

If you wanted to be more explicit, you could say:

• Después de ir a la playa...
• Al volver de la playa...

But después de la playa is completely normal.

It comes from tender.

In this sentence, tender means to hang out, to spread out, or to lay out something, especially clothes or towels so they can dry.

The present tense forms are stem-changing:

• yo tiendo
• tú tiendes
• él/ella tiende
• nosotros tendemos
• vosotros tendéis
• ellos tienden

So tiendo is just the I form of tender.

Because tender is an e > ie stem-changing verb in the present tense.

That means the e in the stem changes to ie in most present forms:

• tender → tiendo
• tender → tiendes
• tender → tiende

But not in nosotros and vosotros:

• tendemos
• tendéis

This is a very common pattern in Spanish, like:

• pensar → pienso
• cerrar → cierro
• empezar → empiezo

In this context, tiendo la toalla means I hang the towel out to dry.

In Spain, tender is very commonly used for hanging laundry, towels, clothes, etc. So it sounds very natural here.

Yes, cuelgo la toalla is also possible, but it focuses more on the act of hanging it. Tiendo la toalla especially suggests hanging it out in order for it to dry, like laundry.

So:

• tender la toalla = hang it out to dry
• colgar la toalla = hang the towel

Both can work, but tender fits this drying context especially well.

Because en shows location: the towel is being hung on / in the balcony area.

• en el balcón = on the balcony / out on the balcony

Using al balcón would suggest movement to the balcony, not the place where the towel is left drying.

So:

• Pongo la silla en el balcón. = I put the chair on the balcony.
• Voy al balcón. = I go to the balcony.

In your sentence, the important idea is the location where the towel is placed, so en is the normal choice.

Spanish often uses the definite article where English would not.

Here, la toalla refers to a specific towel: the towel from the beach, the one already understood from the situation.

Spanish does this a lot with everyday objects:

• Abro la puerta.
• Lavo el coche.
• Recojo la ropa.

In English we might sometimes say I hang up a towel or I hang up the towel, depending on context. In Spanish, la toalla is very natural because the towel is specific and known.

Because para que is used when a new clause follows, and that clause has its own verb.

Here the structure is:

• tiendo la toalla = main action
• para que se seque = purpose clause

The idea is I hang the towel so that it dries.

Spanish usually uses:

• para + infinitive when the subject stays the same
• para que + subjunctive when there is a different subject, or when the second action is presented as a separate clause

Compare:

• Salgo temprano para llegar a tiempo.
  Same subject: I leave early to arrive on time

• Abro la ventana para que entre aire.
  Different subject in the second clause: I open the window so that air comes in

In your sentence, the towel is the thing that ends up drying, so para que se seque sounds very natural.

Because para que is followed by the subjunctive.

This is one of the most common subjunctive triggers in Spanish. It introduces purpose:

• para que descanses
• para que venga
• para que funcione
• para que se seque

The drying is the intended result, not a stated fact. That is why Spanish uses the subjunctive here.

So:

• para que se seque = so that it may dry / so that it dries

You do not use the indicative after para que in standard Spanish.

Here secarse means to dry or to get dry.

So se seque means it dries / it gets dry.

This is a very common kind of Spanish verb where English often uses a simple intransitive verb:

• La ropa se seca al sol. = The clothes dry in the sun.
• Mi pelo se seca rápido. = My hair dries quickly.

The se does not mean the towel is consciously doing something to itself. It is more like a natural change of state: become dry.

So in this sentence:

• secar = to dry something
• secarse = to dry / to become dry

Compare:

• Seco la toalla. = I dry the towel.
• La toalla se seca. = The towel dries.

Because the verb is secar, and in forms before e, Spanish changes c to qu to keep the hard k sound.

So:

• secar → seque
• sacar → saque
• tocar → toque

If Spanish wrote sece, it would sound different. The qu keeps the pronunciation consistent.

This is just a spelling change, not a different verb.

It is not absolutely required, but it is very natural.

Después de la playa is an introductory time phrase, and Spanish often puts a comma after that kind of phrase, especially in writing:

• Después de la playa, tiendo la toalla...
• Por la mañana, salgo a correr.
• Al llegar a casa, me ducho.

Without the comma, the sentence is still understandable:

• Después de la playa tiendo la toalla en el balcón para que se seque.

But the comma helps readability and sounds more polished in writing.