Hän saapui saapumisaulaan ja sanoi matkatavaran jääneen toiseen koneeseen.

Hän means both he and she. Finnish does not normally mark gender in third-person singular pronouns.

So Hän saapui... can mean either:

• He arrived...
• She arrived...

Only context tells you which one is meant.

Saapui is the past tense of saapua, meaning to arrive.

• saapua = to arrive
• saapui = arrived

So Hän saapui means he/she arrived.

Yes. They come from the same root.

• saapua = to arrive
• saapuminen = arriving, arrival
• saapumisaula = arrival hall / arrivals hall

So saapumisaula is basically arrival hall:

• saapumis- = arrival
• aula = hall, lobby

This kind of word-building is very common in Finnish.

Because the sentence needs the form meaning into the arrival hall or to the arrival hall.

Saapumisaulaan is the illative case, which often expresses movement into something.

• saapumisaula = the arrival hall
• saapumisaulaan = into the arrival hall

Since the verb is about arriving somewhere, Finnish uses the destination form.

A useful comparison:

• aulassa = in the hall
• aulaan = into the hall

Finnish often has two ways to say this kind of thing.

A more straightforward version would be:

• Hän saapui saapumisaulaan ja sanoi, että matkatavara oli jäänyt toiseen koneeseen.

Here että means that.

But your sentence uses a more compact structure:

• sanoi matkatavaran jääneen toiseen koneeseen

This is a common written-style construction that corresponds to:

• said that the luggage had been left on another plane
• said that the luggage had remained on another plane

So Finnish does not always need an explicit että in this kind of reported statement.

Because this sentence uses a special reported-speech structure after sanoi.

In the fuller version:

• että matkatavara oli jäänyt...

the subject is matkatavara.

But in the compact version:

• matkatavaran jääneen...

the subject appears in the genitive:

• matkatavara → matkatavaran

So this is not a normal direct object here. It is the subject of that embedded idea, just in a special form required by the construction.

Yes, but matkatavara works like a mass noun, much like English luggage or baggage.

So even if several bags are meant, Finnish can still use the singular:

• matkatavara = luggage, baggage

That is very natural.

If you want to talk about individual items, you might use words like:

• laukku = bag
• matkatavarat = baggage items / luggage, in a more countable sense

But in airport language, matkatavara in the singular is very common.

Jääneen comes from the verb jäädä, which means things like to remain, to be left, to stay behind.

In this sentence, jääneen is part of the compact reported-speech structure after sanoi. It corresponds roughly to had remained / had been left.

So:

• matkatavaran jääneen toiseen koneeseen

corresponds to:

• että matkatavara oli jäänyt toiseen koneeseen

In other words, jääneen here represents the idea had been left behind.

This is a very common thing learners notice.

• toisessa koneessa = in another plane
• toiseen koneeseen = into/on another plane

With jäädä, Finnish often uses the destination-type case when something gets left behind somewhere, especially with vehicles:

• jäädä junaan = to be left on the train
• jäädä bussiin = to be left on the bus
• jäädä koneeseen = to be left on the plane

So even though English might use a static expression like on another plane, Finnish naturally says toiseen koneeseen.

Yes, kone basically means machine, but in travel contexts it very often means airplane.

So at an airport:

• kone = plane
• toiseen koneeseen = onto another plane / on another aircraft

Finnish often relies on context for meanings like this.

It is grammatical and natural, but the structure sanoi matkatavaran jääneen... is more compact and a bit more formal or written-style than everyday spoken Finnish.

In speech, many people would more likely say:

• Hän saapui saapumisaulaan ja sanoi, että matkatavara oli jäänyt toiseen koneeseen.

That version is usually easier for learners too, because it maps more directly onto English said that....