När vi väl var hemma igen märkte jag att det fanns lite skinka kvar, så jag åt en sista smörgås i köket.

Because Swedish is a V2 language in main clauses: the finite verb usually comes in second position.

The sentence starts with the subordinate clause När vi väl var hemma igen. After that whole clause, the main clause begins, and the verb must come before the subject:

• När vi väl var hemma igen märkte jag ...

Compare:

• Jag märkte att ...
• Igår märkte jag att ...
• När vi kom hem märkte jag att ...

So the inversion happens because something other than the subject comes first.

Here väl adds a nuance like once, finally, or after all that.

So När vi väl var hemma igen suggests something like:

• When we were finally back home
• Once we were back home

It often gives a sense that getting home was a step before something else could happen.

This is not the same as the other common väl meaning probably / I suppose / surely. In this sentence, it has the finally/once sense.

Because hemma means at home, while hem means homeward / to home.

• Vi var hemma igen = We were home again / We were back at home
• Vi gick hem igen = We went home again

Since the verb is var (were), the sentence is describing a location/state, not movement. That is why hemma is used.

Igen usually means again.

In hemma igen, it often works idiomatically as back home:

• Vi var hemma igen = We were home again / We were back home

So although igen literally means again, in this kind of context English often prefers back.

Det fanns is the past tense of det finns, which is the normal Swedish way to say there is / there are.

So:

• det finns lite skinka kvar = there is some ham left
• det fanns lite skinka kvar = there was some ham left

The det here is a dummy subject, similar to English there in there was. It does not mean it in the normal sense.

Because skinka here is being used as an uncountable noun, like ham in English.

• lite skinka = a little ham / some ham

You do not need an article before it. This is similar to:

• lite bröd = some bread
• lite mjölk = some milk

If you were talking about a specific countable item, the structure would be different.

Kvar means left, remaining, or still there.

So:

• det fanns lite skinka kvar = there was a little ham left

It is very common in Swedish after nouns or amounts:

• Har du kaffe kvar? = Do you have any coffee left?
• Det är två minuter kvar. = There are two minutes left.

Here så means so, therefore, or and so.

It connects the two ideas:

• there was some ham left
• therefore I ate one last sandwich

So:

• ..., så jag åt en sista smörgås ...
• ..., so I ate one last sandwich ...

This is a very common conversational way to link cause and result.

Åt is the past tense of äta (to eat).

Principal forms:

• äta = infinitive
• äter = present
• åt = past
• ätit = supine/past participle form used in perfect constructions

Examples:

• Jag äter nu. = I am eating now.
• Jag åt då. = I ate then.
• Jag har ätit. = I have eaten.

It is an irregular verb, so this form has to be learned.

En sista smörgås means one last sandwich.

It focuses on the idea of one more, the final one I was going to eat.

• en sista smörgås = one last sandwich

By contrast, den sista smörgåsen means the last sandwich, usually referring to a specific final sandwich in a known set.

• Jag åt en sista smörgås. = I ate one last sandwich.
• Jag åt den sista smörgåsen. = I ate the last sandwich.

In your sentence, the speaker means they decided to eat one final sandwich, not necessarily emphasizing that it was the only sandwich left in the world or in a clearly defined set.

I köket means in the kitchen, and it is a place expression.

Swedish often places time, manner, and place elements fairly late in the clause, especially after the object:

• jag åt en sista smörgås i köket

This sounds natural and neutral. Swedish word order is flexible in some cases, but this version is the most straightforward.

You could move elements around for emphasis, but the given order is the normal one.

Because att introduces a subordinate clause, similar to that in English.

• märkte jag att det fanns lite skinka kvar
• I noticed that there was some ham left

In English, that is often optional:

• I noticed that there was some ham left
• I noticed there was some ham left

In Swedish, att is very commonly included in this structure.

The comma is separating longer parts of the sentence and makes it easier to read.

The structure is roughly:

1. När vi väl var hemma igen
2. märkte jag att det fanns lite skinka kvar
3. så jag åt en sista smörgås i köket

In Swedish, commas are not always used exactly the same way as in English, but before så in a longer sentence like this, a comma is very natural and helpful. It marks the shift to the consequence: so I ate...