De rest van de soep staat nog in de koelkast.

In Dutch, de rest van ... is the normal way to say the rest of ....

So:

• de rest van de soep = the rest of the soup
• de rest van het brood = the rest of the bread
• de rest van de dag = the rest of the day

Using van here is very natural and standard.
De rest soep is not correct Dutch.

Because rest is a de-word in Dutch: de rest.

This is something you mostly have to learn with the noun. There is no special rule here that would make it het.

Examples:

• de rest is klaar = the rest is ready
• de rest komt later = the rest is coming later

Dutch often uses a position verb where English just uses to be.

Here, staan literally means to stand, but in everyday Dutch it is also used for things like containers, bottles, bowls, pans, and other items that are in an upright position somewhere.

So in this sentence, Dutch is really imagining the soup as being in a container that is standing in the fridge.

• De rest van de soep staat nog in de koelkast.
  = The rest of the soup is still in the fridge.

English would normally just say is, but Dutch often prefers staat in this kind of situation.

Yes, that is understandable and grammatically possible.

But:

• is is more neutral
• staat sounds more natural if you are thinking of the soup in a pot, bowl, or container in the fridge

So staat is often the more idiomatic everyday choice.

Dutch makes a distinction that English usually does not:

• staan = standing / upright
• liggen = lying / horizontal
• zitten = being in / contained in / stuck in

If the leftovers are in a bowl, pan, or container in the fridge, Dutch often uses staan.

You would use liggen more for things that are thought of as lying flat or horizontally:

• Het brood ligt in de kast. = The bread is in the cupboard.
• Het boek ligt op tafel. = The book is on the table.

So staat is chosen because the soup is being understood as something in a container that stands in the fridge.

Here nog means still.

So:

• staat nog in de koelkast = is still in the fridge

It suggests that the soup has not been used, eaten, or removed yet.

Compare:

• De soep staat in de koelkast. = The soup is in the fridge.
• De soep staat nog in de koelkast. = The soup is still in the fridge.

Because koelkast is a de-word: de koelkast.

So:

• in de koelkast = in the fridge

This is just something you learn with the noun.

Examples:

• de koelkast
• de tafel
• de fles

Because the subject is de rest van de soep, and the grammatical head of that subject is de rest.

De rest is singular, so the verb is singular too:

• De rest staat ...
• not De rest staan ...

This is the same idea as in English:

• The rest is ...
• not The rest are ...
  though English can sometimes vary depending on meaning

In this Dutch sentence, singular agreement is the normal choice.

Dutch main clauses usually follow the verb-second rule.

The sentence is:

• De rest van de soep = subject
• staat = finite verb
• nog = adverb
• in de koelkast = place phrase

So the structure is:

subject + finite verb + rest

That is why you get:

De rest van de soep staat nog in de koelkast.

If you move another element to the front, the verb still stays in second position:

• In de koelkast staat de rest van de soep nog.

That version is possible, though the original sentence is more neutral.

Van de here means of the.

So:

• van = of
• de = the

Together:

• de rest van de soep = the rest of the soup

This is a very common Dutch pattern:

• het einde van de film = the end of the film
• de kleur van de auto = the color of the car
• de naam van de stad = the name of the city

Grammatically it says the rest of the soup, but the use of staat shows that Dutch is thinking in a practical, physical way about where it is: probably in a pot, bowl, or container that is standing in the fridge.

This is very normal in Dutch. The language often chooses the verb based on the physical position of the object or container, even when English would simply say is.

So the meaning is still about the soup, but the verb reflects the way it is stored.