Kuŝante sur la sofo, ŝi diras, ke ŝia stomako doloras.

Kuŝante is the adverbial present participle of kuŝi.

• kuŝi = to lie, recline
• -ant- = present active participle
• -e = adverb ending

So kuŝante means while lying, lying, or as she lies.

Also, kuŝi means physically lying down or reclining, not telling a lie.

Because the sentence is showing a background action happening at the same time as the main action.

• ŝi diras = the main action
• kuŝante sur la sofo = the circumstance in which she says it

So kuŝante works like while lying on the sofa.

If you used kuŝas, you would need a fuller structure, such as:

• Dum ŝi kuŝas sur la sofo, ŝi diras...

The participle makes the sentence more compact.

It ends in -e because here it is being used adverbially: it describes the situation of the action diras.

• kuŝante = lying, while lying
• kuŝanta = lying, but as an adjective

So:

• kuŝante sur la sofo, ŝi diras... = While lying on the sofa, she says...
• la knabino kuŝanta sur la sofo = the girl lying on the sofa

In this sentence, kuŝante is the most natural form because it describes the manner or circumstance of the main verb.

It refers to ŝi.

In Esperanto, an opening participial phrase like this normally has the same subject as the main clause. So the person who is lying on the sofa is the same person who is speaking.

That means the sentence is understood as:

• She, while lying on the sofa, says...

If someone else were the one lying on the sofa, Esperanto would normally make that explicit with a different structure.

Because this is location, not movement toward a place.

• sur la sofo = on the sofa
• sur la sofon = onto the sofa

Esperanto often uses -n after a preposition to show direction. Here she is already on the sofa, so there is no directional -n.

Compare:

• Ŝi kuŝas sur la sofo. = She is lying on the sofa.
• Ŝi sidiĝas sur la sofon. = She sits down onto the sofa.

La is the definite article, meaning the.

Esperanto has:

• la = the
• no indefinite article for a/an

So sur la sofo means on the sofa, usually a specific sofa understood from the situation.

Using la here is completely normal. In a concrete scene like this, Esperanto usually uses la just as English would use the.

Ke introduces a subordinate clause, like English that.

So:

• ŝi diras, ke... = she says that...

The clause after ke gives the content of what she says:

• ke ŝia stomako doloras

In English, that is often omitted, but in Esperanto ke is normally stated.

Because sia only refers back to the subject of its own clause.

In the clause:

• ke ŝia stomako doloras

the grammatical subject is stomako, not ŝi.

So sia stomako would incorrectly try to refer back to stomako itself. That is why Esperanto uses ŝia here.

A useful comparison:

• Ŝi lavas sian aŭton. = She washes her own car.
  Here ŝi is the subject of the clause, so sian is correct.

But in your sentence:

• Ŝi diras, ke ŝia stomako doloras.

Inside the ke-clause, the subject is stomako, so ŝia is the right choice.

Dolori can work like to hurt or to ache.

In this sentence, stomako is the subject:

• ŝia stomako doloras = her stomach hurts / her stomach aches

So the body part itself is what hurts.

This is a very normal Esperanto pattern, similar to:

• Mia kapo doloras. = My head hurts.
• Liaj dentoj doloras. = His teeth hurt.

You do not need a direct object here.

The commas separate parts of the sentence for clarity.

• Kuŝante sur la sofo,
  The first comma marks off the introductory participial phrase.

• ŝi diras, ke...
  The second comma introduces the ke-clause.

The comma before ke is very common in Esperanto. The comma after the opening phrase is also natural and helps readability.

Yes, Esperanto word order is fairly flexible, as long as the structure stays clear.

For example, you could also say:

• Ŝi diras, ke ŝia stomako doloras, kuŝante sur la sofo.

But the original version is clearer and more natural if you want to set the scene first:

• Kuŝante sur la sofo, ŝi diras, ke ŝia stomako doloras.

Putting kuŝante sur la sofo first immediately gives the listener the context.