Nach dem Waschen ist die Bettwäsche wieder sauber.

Because nach is a preposition that takes the dative case when it means after in a time sense.

• nominative: das Waschen
• dative: dem Waschen

So:

• nach dem Waschen = after the washing / after washing

This is a very common pattern in German:

• nach dem Essen = after eating / after the meal
• nach der Arbeit = after work
• nach dem Unterricht = after class

Because here Waschen is being used as a noun, not as a verb.

The verb is:

• waschen = to wash

But German can turn verbs into nouns. When that happens, they are capitalized:

• das Waschen = the washing
• das Essen = eating / the meal
• das Lesen = reading

In this sentence, dem Waschen comes from das Waschen.

Here das Waschen means the washing or more naturally in English washing.

So nach dem Waschen can be understood as:

• after washing
• after the washing
• once it has been washed

It refers to the washing process, not to a person doing the action in the sentence.

German main clauses follow the verb-second rule: the finite verb must be in the second position.

The sentence begins with the time phrase:

• Nach dem Waschen

That whole phrase counts as the first element. So the finite verb must come next:

• Nach dem Waschen | ist | die Bettwäsche wieder sauber.

If you start with the subject instead, you get:

• Die Bettwäsche ist nach dem Waschen wieder sauber.

Both are grammatical, but the original sentence emphasizes after washing first.

Because in German compounds, the gender is determined by the last part of the compound noun.

• das Bett = bed
• die Wäsche = laundry / washing / linen
• die Bettwäsche = bed linen / bedding

Since Wäsche is feminine, Bettwäsche is also feminine:

• die Bettwäsche

This is a very important rule for compound nouns in German.

It is singular here.

Even though Bettwäsche refers to a set of items like sheets, pillowcases, etc., German often treats it as a collective singular noun, similar to how English can use bedding.

That is why the sentence has:

• die Bettwäsche ist ... not
• die Bettwäsche sind ...

So grammatically, it behaves like a singular noun.

Because sauber is a predicate adjective here, used after the verb sein.

In German, adjectives usually do not take endings when they come after verbs like:

• sein = to be
• werden = to become
• bleiben = to remain

So:

• Die Bettwäsche ist sauber.
• Das Zimmer ist groß.
• Die Kinder sind müde.

Adjective endings only appear when the adjective comes before a noun, for example:

• die saubere Bettwäsche = the clean bed linen

Here wieder means again or once again, but in natural English it often comes across as clean again or clean once more.

So the idea is:

• before washing, it was not clean
• after washing, it is clean again

In this sentence, wieder modifies the adjective sauber in meaning:

• wieder sauber = clean again

Because wieder is an adverb, and in this sentence it naturally comes before the adjective it helps describe:

• wieder sauber = clean again

This is the normal word order in German.

Compare:

• Die Bettwäsche ist wieder sauber. = The bed linen is clean again.
• Die Bettwäsche ist sauber. = The bed linen is clean.

Adding wieder shows a return to a previous state.

Yes. That is also correct.

German is flexible about word order as long as the verb-second rule is respected in main clauses.

Compare:

• Nach dem Waschen ist die Bettwäsche wieder sauber.
• Die Bettwäsche ist nach dem Waschen wieder sauber.

The difference is mainly emphasis:

• starting with Nach dem Waschen emphasizes the time point
• starting with Die Bettwäsche emphasizes the subject

They are related in meaning, but they are used differently.

nach dem Waschen

• preposition + noun
• literally after the washing
• compact and noun-based

nachdem

• conjunction
• introduces a subordinate clause
• means after in the sense of after something happened

For example:

• Nach dem Waschen ist die Bettwäsche wieder sauber.
• Nachdem die Bettwäsche gewaschen wurde, ist sie wieder sauber.

The first version is shorter and more natural in everyday German here.

Sometimes German can omit the article with certain abstract or general nouns, but with Bettwäsche, using die is very natural here.

• Die Bettwäsche ist wieder sauber. = the bed linen is clean again

Without the article:

• Bettwäsche ist wieder sauber

This sounds less natural in this context unless you are speaking in a very general or headline-like style. For an ordinary sentence, die Bettwäsche is the safer and more idiomatic choice.

A helpful breakdown is:

• Nach dem Waschen = after washing
• ist = is
• die Bettwäsche = the bed linen / bedding
• wieder sauber = clean again

So the grammar is:

• time phrase
  • verb
    • subject
      • adverb + adjective

That gives:

• Nach dem Waschen ist die Bettwäsche wieder sauber.

A very natural English equivalent is:

• After washing, the bed linen is clean again.