Nach dem Frühstück wische ich den Tisch mit einem Schwamm ab.

Because nach takes the dative case in this meaning.

• das Frühstück is the base form.
• In the dative, das becomes dem.
• So: nach dem Frühstück = after breakfast

This is a very common pattern:

• nach dem Essen = after the meal / after eating
• nach der Schule = after school
• nach dem Film = after the film

So the important rule is: nach + dative.

German main clauses follow the verb-second (V2) rule. That means the finite verb must be the second element in the sentence.

In this sentence, Nach dem Frühstück comes first, so the verb wische must come next:

• Nach dem Frühstück | wische | ich ...

If you started with ich, that would also be correct:

• Ich wische nach dem Frühstück den Tisch mit einem Schwamm ab.

Both are grammatical. The original sentence just puts the time expression first for emphasis or flow.

For the same V2 reason. Since Nach dem Frühstück is in the first position, the verb has to come immediately after it.

Compare:

• Ich wische den Tisch ab.
• Nach dem Frühstück wische ich den Tisch ab.

In English, word order changes less. In German, when you move another element to the front, the verb still stays in second position, so the subject often moves after the verb.

Because den Tisch is the direct object, so it is in the accusative case.

The noun Tisch is masculine:

• nominative: der Tisch
• accusative: den Tisch

Here, the thing being wiped is the table, so it is the direct object:

• Ich wische den Tisch ab.

That is why you need den, not der.

Because mit always takes the dative case.

The noun Schwamm is masculine:

• nominative: ein Schwamm
• dative: einem Schwamm

So:

• mit einem Schwamm = with a sponge

This is another very useful rule to memorize:

• mit dem Auto = by car / with the car
• mit meiner Schwester = with my sister
• mit einem Messer = with a knife

So here, mit + dative gives mit einem Schwamm.

ab is the separable prefix of the verb abwischen.

• wischen = to wipe
• abwischen = to wipe off / wipe down

In a normal main clause, separable prefixes move to the end:

• Ich wische den Tisch ab.
• Er räumt das Zimmer auf.
• Sie macht das Licht aus.

So the full verb is really abwischen, but in this sentence it gets split:

• wische ... ab

That is why ab appears at the end.

wischen can mean to wipe in a more general sense.

abwischen usually has the idea of wiping something off, wiping something clean, or wiping down a surface.

So in this sentence, abwischen is very natural because you are wiping the table as a surface.

Roughly:

• wischen = wipe
• abwischen = wipe off / wipe down

In everyday German, den Tisch abwischen is a very common phrase.

Yes. It tells you when the action happens.

So the sentence structure is roughly:

• Nach dem Frühstück = time
• wische ich = verb + subject
• den Tisch = direct object
• mit einem Schwamm = how / with what
• ab = separable prefix

A learner can think of the sentence as:

After breakfast, I wipe the table down with a sponge.

Yes. German word order is flexible, as long as the grammar rules are respected.

For example, these are also possible:

• Ich wische nach dem Frühstück den Tisch mit einem Schwamm ab.
• Den Tisch wische ich nach dem Frühstück mit einem Schwamm ab.
• Mit einem Schwamm wische ich nach dem Frühstück den Tisch ab.

These versions can sound more or less natural depending on context, but they are grammatically possible.

The main rule to remember is:

• in a main clause, the finite verb still has to be in second position
• the separable prefix ab still goes to the end

Because Frühstück is a noun, and all nouns are capitalized in German.

Examples:

• das Frühstück = breakfast
• der Tisch = table
• der Schwamm = sponge

This is one of the most noticeable spelling rules in German: all nouns start with a capital letter.

Here it is definitely a noun: das Frühstück = breakfast.

There is also the verb frühstücken, which means to have breakfast.

Compare:

• Nach dem Frühstück wische ich den Tisch ab.
  = After breakfast, I wipe the table.

• Ich frühstücke um sieben.
  = I have breakfast at seven.

So the sentence uses the noun form, not the verb.

German often uses an article where English does not.

In English, after breakfast sounds natural without the. In German, the usual expression is:

• nach dem Frühstück

Literally this looks like after the breakfast, but you should not translate it too literally. It is simply the normal German way to say after breakfast in this context.

This is something learners often have to get used to: German and English do not always use articles in the same places.

Yes, it can.

For example:

• Nach dem Frühstück wische ich mit einem Schwamm den Tisch ab.
• Ich wische den Tisch mit einem Schwamm ab.

Both are possible.

However, den Tisch mit einem Schwamm abwischen is a very natural order because:

• den Tisch comes early as the direct object
• mit einem Schwamm adds the instrument afterward

So while German allows flexibility, the original sentence sounds very normal and idiomatic.