Meine Schwester stellt den Wäscheständer auf den Balkon, damit der Kissenbezug schneller trocknet.

Because aufstellen is a separable verb in German.

• Infinitive: aufstellen = to set up / put up / place upright
• In a main clause, it splits:
  • stellt goes in the normal verb position
  • auf goes to the end

So:

• Meine Schwester stellt den Wäscheständer auf den Balkon auf.

However, in your sentence, the object den Wäscheständer comes before auf den Balkon, so the sentence is:

• Meine Schwester stellt den Wäscheständer auf den Balkon, damit ...

Many learners expect a final auf, but here auf den Balkon stellen is often understood as the full directional phrase with stellen, and in natural usage the structure can feel streamlined because auf den Balkon stellen already clearly expresses the action of placing it there. In teaching contexts, learners should mainly remember that aufstellen is separable and that separable prefixes normally go to the end in main clauses.

Because Wäscheständer is the direct object of the verb, so it is in the accusative case.

• der Wäscheständer = nominative
• den Wäscheständer = accusative

Here, your sister is doing something to the drying rack: she is placing it somewhere. That makes it the direct object.

Because this sentence expresses movement toward a destination, not location.

German two-way prepositions like auf can take:

• accusative for direction/movement
• dative for location

So:

• auf den Balkon = onto / to the balcony
• auf dem Balkon = on the balcony

Here the drying rack is being moved to the balcony, so German uses accusative:

• auf den Balkon

If it were already there, you would say:

• Der Wäscheständer steht auf dem Balkon.

Because der Kissenbezug is the subject of the clause damit der Kissenbezug schneller trocknet.

In that clause, the pillowcase is the thing doing the drying, so it is nominative:

• der Kissenbezug = nominative subject

Compare:

• Meine Schwester wäscht den Kissenbezug.
  Here it is the object, so: den Kissenbezug
• Der Kissenbezug trocknet.
  Here it is the subject, so: der Kissenbezug

Here damit means so that or in order that.

It introduces a subordinate clause, and in German subordinate clauses, the conjugated verb goes to the end.

So:

• ..., damit der Kissenbezug schneller trocknet.

Word order inside that clause:

1. damit
2. der Kissenbezug
3. schneller
4. trocknet

This is very normal German subordinate-clause word order.

Because German often uses trocknen directly, just like English to dry.

• etwas trocknet = something dries
• etwas trocknen can also mean to dry something, depending on context

Here:

• der Kissenbezug trocknet = the pillowcase dries / is drying

German does not need a phrase like becomes dry here.

Here schneller is an adverb/comparative form, meaning faster or more quickly.

It is not describing a noun, so it does not take adjective endings.

• ein schneller Zug = a fast train
  (schneller is an adjective modifying Zug)
• Der Zug fährt schneller. = The train goes faster
  (schneller is being used adverbially)

In your sentence:

• damit der Kissenbezug schneller trocknet
  = so that the pillowcase dries faster

It is a compound noun:

• Wäsche = laundry
• Ständer = stand / rack

So Wäscheständer literally means something like laundry stand, but the natural English meaning is drying rack or clothes horse.

German uses compound nouns very often, and the last part usually determines the grammatical gender. Since der Ständer is masculine, der Wäscheständer is also masculine.

Because meine Schwester is the subject of the main clause, so it is in the nominative case.

• meine Schwester = nominative
• meiner Schwester = dative or genitive, depending on context

Here:

• Meine Schwester stellt ...
  = My sister puts ...

Since she is the one performing the action, nominative is required.

Not naturally in this exact sentence, because um ... zu is usually used when the subject of both clauses is the same.

Here the subjects are different:

• Main clause subject: Meine Schwester
• Subordinate clause subject: der Kissenbezug

So German uses damit:

• Meine Schwester stellt den Wäscheständer auf den Balkon, damit der Kissenbezug schneller trocknet.

If the subject were the same, um ... zu would work better. For example:

• Meine Schwester geht auf den Balkon, um die Wäsche aufzuhängen.

It simply means the pillowcase in the singular. The sentence is talking about one pillowcase.

• singular: der Kissenbezug
• plural: die Kissenbezüge

If there were several pillowcases, the clause would be:

• ..., damit die Kissenbezüge schneller trocknen.

Notice that both the article and the verb would change:

• die Kissenbezüge = plural subject
• trocknen = plural verb

These verbs all involve putting something somewhere, but German often distinguishes them by position:

• stellen = put something in a standing/upright position
• legen = lay something down
• setzen = set someone/something into a sitting position

A Wäscheständer normally stands upright, so stellen is the natural verb:

• den Wäscheständer auf den Balkon stellen

This is one of those very German patterns that does not always match English exactly, since English often just uses put for all of them.