Nadat wij hebben gegeten, ruimt Tom de borden op en zet hij alles in de vaatwasser.

Nadat means after when it introduces a whole clause with its own subject and verb.

• Nadat wij hebben gegeten = after we have eaten
• Na het eten = after the meal / after eating

So:

• na
  • noun phrase
• nadat
  • full clause

That is why the sentence uses nadat and not na.

This is the present perfect:

• hebben = auxiliary verb
• gegeten = past participle of eten

Because the subject is wij (we), the auxiliary must be hebben.

So:

• ik heb gegeten
• jij hebt gegeten
• wij hebben gegeten

In this sentence, hebben gegeten means have eaten.

Because nadat introduces a subordinate clause. In Dutch subordinate clauses, the verbs typically move toward the end of the clause.

So instead of something like:

• wij hebben gegeten in a main-clause pattern,

you get a subordinate-clause structure after nadat:

• Nadat wij hebben gegeten

This is very normal Dutch word order after conjunctions like omdat, terwijl, als, voordat, and nadat.

This is the Dutch verb-second rule.

The clause Nadat wij hebben gegeten comes first, and it takes up the first position in the sentence. In a Dutch main clause, the finite verb must then come in the second position.

So the order becomes:

• Nadat wij hebben gegeten, ruimt Tom ...

not:

• Nadat wij hebben gegeten, Tom ruimt ...

This is called inversion: the verb comes before the subject.

Because the sentence begins with a subordinate clause:

• Nadat wij hebben gegeten

Dutch normally separates that introductory clause from the main clause with a comma. It works much like English punctuation in sentences such as After we have eaten, Tom...

Because opruimen is a separable verb.

The full verb is:

• opruimen = to clear away / tidy up

In a main clause with a finite verb, Dutch often splits separable verbs:

• Tom ruimt de borden op

So:

• ruimt = the conjugated verb part
• op = the separated prefix

But in the dictionary form or infinitive, it stays together:

• opruimen

Because the initial time clause still applies to both actions:

• Nadat wij hebben gegeten, ruimt Tom de borden op
• en zet hij alles in de vaatwasser

Dutch often keeps the same inverted pattern in the second coordinated clause, so zet hij matches ruimt Tom.

This makes the two actions feel parallel:

• first he clears the plates away
• then he puts everything in the dishwasher

Both wij and we mean we.

The difference is mainly style and emphasis:

• we is more common in everyday speech
• wij sounds a bit fuller, clearer, or slightly more formal/emphatic

So Nadat wij hebben gegeten and Nadat we hebben gegeten are both possible. The version with wij is just a bit more explicit.

Dutch uses in both for in and often for English into when the verb already shows movement or placement.

So:

• het staat in de vaatwasser = it is in the dishwasher
• hij zet het in de vaatwasser = he puts it into the dishwasher

The movement is understood from zet. Dutch does not need a separate word exactly like English into here.

Dutch often uses a placement verb for putting something somewhere.

Here, zetten is very natural in:

• iets in de vaatwasser zetten

It means putting something into its place, in this case into the dishwasher. In casual speech, some people might also say doet ... in de vaatwasser, but zet ... in de vaatwasser is a very normal and idiomatic choice.

Dutch often uses the present tense for:

• habitual actions
• routines
• planned or expected future actions

So this sentence can mean something like a routine:

• after we eat, Tom clears the plates and puts everything in the dishwasher

It does not have to mean that it is happening right this second. Dutch present tense is broader than English learners sometimes expect.

Alles means everything.

In this context, it probably refers to all the meal-related items, not literally every object in the room. So it means something like:

• the plates
• cutlery
• cups or glasses
• other dishes

It is a convenient Dutch word for all of it / everything.