Han gikk hjem uten at han sa noe.

Because the sentence is in the past tense.

• går = go / is going
• gikk = went

So Han gikk hjem means He went home.

In Norwegian, hjem often works like the English adverb home.

• Han gikk hjem = He went home

You do not usually say til hjem.
That would sound unnatural, just like went to home sounds unnatural in English.

Compare:

• Han gikk hjem = He went home
• Han gikk til huset = He walked to the house

Uten at means without followed by a whole clause.

So:

• uten at han sa noe = without him saying anything / without saying anything

This is a very common Norwegian structure when English might use without + -ing.

Examples:

• Hun dro uten at hun sa farvel. = She left without saying goodbye.
• De gikk uten at noen merket det. = They left without anyone noticing.

Because at introduces a subordinate clause.

In uten at han sa noe, the part after at has its own subject and verb:

• han = subject
• sa = verb

So literally, Norwegian is structuring it as:

• without that he said anything

That literal English version is not natural English, but it helps explain the grammar.

Because after at, Norwegian normally uses subordinate clause word order, where the subject comes before the verb.

So:

• uten at han sa noe = correct
• uten at sa han noe = incorrect

This is different from main clauses, where Norwegian often has verb-second word order:

• Han sa noe.
• I går sa han noe.

But after at, the subject normally stays before the verb:

• ... at han sa noe

After a negative-type expression like uten (without), Norwegian usually uses noe rather than ingenting.

So:

• uten at han sa noe = without saying anything

This is similar to English, where anything is more natural than nothing in this kind of sentence.

Compare:

• Han sa ikke noe. = He didn’t say anything.
• Han sa ingenting. = He said nothing.

Both can be correct in some contexts, but in this sentence noe is the natural choice.

Yes, and that is often the most natural alternative.

• Han gikk hjem uten å si noe.
• Han gikk hjem uten at han sa noe.

Both mean roughly the same thing: He went home without saying anything.

The difference is mainly structure:

• uten å si noe = without saying anything
• uten at han sa noe = without him saying anything / without he said anything

The uten å + infinitive version is often used when the subject is the same as in the main clause.

Yes, a small one.

• uten å si noe is usually simpler and more direct.
• uten at han sa noe can sound a bit more explicit or slightly more formal, because it names the subject inside the subordinate clause.

Since the subject is the same in both parts of your sentence, many speakers would probably prefer:

• Han gikk hjem uten å si noe.

But uten at han sa noe is still perfectly correct.

Because si means say, while snakke means speak / talk.

• sa noe = said something / said anything
• snakket = talked / spoke

In this sentence, the idea is that he did not say anything, so sa noe fits better.

Compare:

• Han sa noe. = He said something.
• Han snakket med meg. = He talked to me.

Yes. Noe can mean both something and anything, depending on the context.

Examples:

• Han sa noe. = He said something.
• Han sa ikke noe. = He didn’t say anything.
• uten at han sa noe = without saying anything

So you have to understand it from the sentence around it.

Yes, it is grammatical and understandable.

However, many speakers would probably more naturally say:

• Han gikk hjem uten å si noe.

That version is shorter and very common.
Your original sentence is still correct, but it is a bit heavier because it uses a full clause after uten at.

Sometimes, yes.

gå / gikk can mean:

• go / went
• walk / walked

So Han gikk hjem can mean:

• He went home
• He walked home

Which meaning is intended depends on context. If no context is given, English often translates it simply as He went home.

Because the second part is a separate clause, and that clause needs its own subject.

Main clause:

• Han gikk hjem

Subordinate clause:

• han sa noe

So Norwegian repeats han because the clause after uten at is complete on its own.

In English, we often avoid this by using without saying anything, but Norwegian can keep the full clause structure.