Efter middagen står tallrikarna kvar på diskbänken.

Because Swedish main clauses follow the V2 rule: the finite verb must come in the second position.

Here, Efter middagen is placed first, so the verb står has to come next:

• Efter middagen = first position
• står = second position
• tallrikarna = subject, after the verb

So this pattern is normal:

• Efter middagen står tallrikarna kvar på diskbänken.
• I dag kommer hon inte.
• På bordet ligger boken.

If the subject comes first, then the verb comes after it:

• Tallrikarna står kvar på diskbänken efter middagen.

Both are correct, but the first version emphasizes after dinner a bit more.

Because Swedish often uses the definite form for meals when talking about a specific meal in the day:

• efter frukosten = after breakfast
• efter lunchen = after lunch
• efter middagen = after dinner

So middagen means the dinner / dinner in the sense of the meal that has just happened.

For an English speaker, this can feel strange, because English usually says just after dinner without the. Swedish often prefers the definite form here.

Tallrikarna is the definite plural form of tallrik.

Here is the pattern:

• en tallrik = a plate
• tallriken = the plate
• tallrikar = plates
• tallrikarna = the plates

So Swedish usually puts the definite article at the end of the noun instead of using a separate word like English the.

Swedish very often uses a position verb for location, where English would simply use is/are.

So instead of saying the plates are on the counter, Swedish prefers a verb like:

• står = stand
• ligger = lie
• sitter = sit

In this sentence, står is being used to describe the plates as being located there in a natural, everyday way.

Using är would usually sound less idiomatic if you are talking about where physical objects are placed.

Sometimes, yes. It depends on how you imagine the plates.

• står often suggests things that are set/placed somewhere, especially objects on a surface
• ligger suggests something is lying flat

So if you picture plates lying flat on the counter, ligger could make sense. But står is still very common in everyday Swedish when talking about dishes left somewhere.

So:

• Tallrikarna står kvar på diskbänken sounds very natural
• Tallrikarna ligger kvar på diskbänken is also possible if you want to focus on them lying there flat

This is one of those areas where Swedish often chooses a more specific location verb than English.

Kvar means left, remaining, or still there.

So står kvar means something like:

• are still standing there
• remain there
• have been left there

Without kvar, the sentence would mainly tell you where the plates are:

• Tallrikarna står på diskbänken = The plates are on the counter.

With kvar, it adds the idea that they have not been removed:

• Tallrikarna står kvar på diskbänken = The plates are still left on the counter.

Because på means on, and a diskbänk is a kind of surface/counter.

So:

• på diskbänken = on the counter / on the sink counter

Swedish uses på for things resting on top of a surface.

A useful contrast:

• på diskbänken = on the counter by the sink
• i diskhon = in the sink

So if the plates are actually in the sink, Swedish would usually use i, not på.

Diskbänk usually means the counter/worktop by the sink, often the place where dishes are put before or after washing.

It is not exactly the same as sink.

So:

• diskbänken = the counter by the sink
• diskhon = the sink basin

That distinction is useful, because English speakers may first assume diskbänk means the sink itself.

Yes. For example:

• Tallrikarna står kvar på diskbänken efter middagen.

This is also correct.

The difference is mainly one of focus:

• Efter middagen står tallrikarna kvar på diskbänken.
  This highlights after dinner.

• Tallrikarna står kvar på diskbänken efter middagen.
  This starts more neutrally with the plates.

But in a main clause, Swedish still follows the V2 rule, so if you put Efter middagen first, the verb must come right after it.