Meine Tochter hängt die nassen Socken mit einer Wäscheklammer auf.

Because the verb here is aufhängen, which is a separable verb. In a normal main clause, the conjugated part goes in the usual verb position, and the prefix moves to the end:

• aufhängen = to hang up
• Meine Tochter hängt ... auf.

So German is literally doing something like:

• My daughter hangs ... up.

That is normal with separable verbs such as:

• anrufen → Ich rufe dich an.
• einkaufen → Wir kaufen heute ein.

Because the subject is Meine Tochter = my daughter, which is third person singular. The verb has to match that subject.

The infinitive is hängen, but in the present tense:

• ich hänge
• du hängst
• er/sie/es hängt
• wir hängen
• ihr hängt
• sie/Sie hängen

Since Tochter is singular, you need hängt.

Meine is a possessive determiner meaning my. Its ending changes depending on gender, number, and case.

Here, Tochter is:

• feminine
• singular
• nominative (because it is the subject)

So you get:

• meine Tochter = my daughter

Compare:

• mein Sohn = my son
• meine Tochter = my daughter
• mein Kind = my child
• meine Kinder = my children

This is the direct object of the sentence: the thing being hung up.

The noun Socken is plural, and in the accusative plural, the definite article is still die.

So:

• nominative plural: die Socken
• accusative plural: die Socken

The adjective nass also needs an ending. After die in the plural accusative, it becomes:

• die nassen Socken = the wet socks

So the whole phrase is built like this:

• die = the
• nassen = wet
• Socken = socks

Because den is not the accusative plural article.

For plural nouns, the definite article is:

• nominative plural: die
• accusative plural: die
• dative plural: den
• genitive plural: der

So in this sentence, Socken is plural and accusative, so it must be:

• die Socken

You would use den Socken only in the dative plural, for example:

• mit den Socken = with the socks

Because German adjectives usually change their endings when they come before a noun.

Here, nass comes before Socken, and the phrase is:

• die nassen Socken

The ending -en appears because the adjective follows a definite article and the noun phrase is plural accusative.

A few comparisons:

• der nasse Socken — not correct for standard singular/plural meaning here
• die nasse Socke = the wet sock
• die nassen Socken = the wet socks

So the ending is there because German marks grammar on adjectives too.

Because the preposition mit always takes the dative case.

So after mit, the article has to be dative. Wäscheklammer is a feminine noun, so:

• nominative: eine Wäscheklammer
• accusative: eine Wäscheklammer
• dative: einer Wäscheklammer

That is why the sentence says:

• mit einer Wäscheklammer = with a clothespin

Because the sentence says she is hanging up the socks with a clothespin—that is, using one clothespin as the tool or means.

German often uses singular or plural here depending on the situation:

• mit einer Wäscheklammer = with a clothespin
• mit Wäscheklammern = with clothespins

So the singular is grammatically fine. It just describes the action with one clothespin.

It is a prepositional phrase telling you how or with what the action is done.

The sentence parts are:

• Meine Tochter = subject
• hängt ... auf = verb
• die nassen Socken = direct object
• mit einer Wäscheklammer = extra information: with a clothespin

So it answers a question like:

• How is she hanging them up?
• With a clothespin.

Yes, to some extent. German word order is flexible, but the verb pattern still matters:

• the conjugated verb stays in second position in a main clause
• the separable prefix auf stays at the end

So these are possible:

• Meine Tochter hängt die nassen Socken mit einer Wäscheklammer auf.
• Meine Tochter hängt mit einer Wäscheklammer die nassen Socken auf.
• Die nassen Socken hängt meine Tochter mit einer Wäscheklammer auf.

They all work, though the first one is the most neutral and natural for learners.

Hängen by itself often means to hang or to be hanging, depending on context.

Aufhängen specifically means to hang up.

So:

• Die Socken hängen draußen. = The socks are hanging outside.
• Meine Tochter hängt die Socken auf. = My daughter hangs up the socks.

In this sentence, aufhängen is the better verb because the idea is that she is putting them up to dry.

Yes. Hängt is present tense.

German present tense can cover both:

• My daughter hangs up the wet socks with a clothespin.
• My daughter is hanging up the wet socks with a clothespin.

German usually does not need a separate form like English is hanging. The context tells you whether it is a general action or something happening right now.

A native speaker would understand it very naturally as:

• My daughter is hanging up the wet socks using a clothespin.

The main grammatical ideas are:

• Meine Tochter = subject
• hängt ... auf = separable verb, present tense
• die nassen Socken = direct object, accusative plural
• mit einer Wäscheklammer = dative phrase after mit

So it is a very useful example because it shows separable verbs, case after a preposition, and adjective endings all in one sentence.