Kantarellen og steinsoppen ligger i soppkurven.

Because they are in the definite singular form.

These words follow the en / -en pattern:

• en kantarell → kantarellen
• en steinsopp → steinsoppen
• en soppkurv → soppkurven

So the ending -en is doing the job that the often does in English.

In Norwegian, definiteness is usually added to the end of the noun instead of using a separate article before it.

So:

• English: the chanterelle
• Norwegian: kantarellen

This is very common in Norwegian.
A separate article often appears only in some structures, especially with adjectives, for example:

• den store soppkurven = the big mushroom basket

So in your sentence, the nouns are definite just because of their endings.

Ligger is the present tense of ligge.

In this sentence, it means something like:

• is lying
• is located
• is in

Norwegian often uses position verbs where English would simply use be.

Very commonly:

• ligge = lie, be lying
• stå = stand, be standing
• sitte = sit, be sitting

For mushrooms in a basket, ligger sounds natural because they are thought of as resting there.

Because Norwegian often prefers a more specific position verb.

English often says:

• The mushrooms are in the basket

But Norwegian frequently chooses a verb that describes how something is positioned:

• ligger if it is lying/resting
• står if it is standing upright
• sitter for seated beings, and sometimes for things attached or placed in a certain way

So er i soppkurven is possible in some contexts, but ligger i soppkurven is more natural here.

Because Norwegian verbs do not change form for singular and plural in the present tense.

So you get:

• jeg ligger
• du ligger
• han ligger
• de ligger

The subject can be one thing or several things, but the verb form stays the same.

That is why kantarellen og steinsoppen ligger is correct.

Because Norwegian usually writes compound nouns as one word.

Here:

• sopp = mushroom
• kurv = basket
• soppkurv = mushroom basket
• soppkurven = the mushroom basket

This is very normal in Norwegian.
English often uses two words where Norwegian uses one compound noun.

Because i means in / inside, while på means on / on top of.

A basket is a container, so if the mushrooms are inside it, Norwegian uses i:

• i soppkurven = in the mushroom basket

If something were physically on top of the basket, then på would make sense.

In Norwegian, each noun is normally marked separately.

So if you mean two specific things, you say:

• kantarellen og steinsoppen

Not just one definite ending for both.

Compare:

• kantarellen og steinsoppen = the chanterelle and the porcini
• kantarell og steinsoppen = a chanterelle and the porcini

So if only one noun is definite, the meaning changes.

It follows a very common Norwegian pattern:

• Subject: Kantarellen og steinsoppen
• Verb: ligger
• Prepositional phrase: i soppkurven

So it is basically:

subject + verb + place

This is a normal main-clause word order in Norwegian.

You can also move the place phrase to the front:

• I soppkurven ligger kantarellen og steinsoppen.

That is also correct, but then the verb still stays in the second position, which is an important Norwegian rule.

It talks about specific mushrooms, not the kinds in general.

That is because both nouns are definite singular:

• kantarellen = the chanterelle
• steinsoppen = the porcini

So the sentence refers to particular items, probably ones already known from the context.

If you wanted a more general statement, you would usually use a different form, for example plural or indefinite forms depending on the meaning.