Door die nieuwe trui begonnen mijn armen te jeuken, en ik zag Tom ook zweten in de volle bus.

Door die nieuwe trui means because of that new sweater.

Dutch often lets you begin a sentence with the cause, time, place, or another extra piece of information. When that happens, the finite verb usually comes next. That is why you get:

• Door die nieuwe trui begonnen mijn armen te jeuken
• literally: Because of that new sweater began my arms to itch

In more neutral order, it would be:

• Mijn armen begonnen door die nieuwe trui te jeuken.

Both are correct, but starting with Door die nieuwe trui gives that cause a bit more emphasis.

This is because of the verb-second rule in Dutch main clauses.

In a normal main clause, the finite verb is in the second position:

• Mijn armen begonnen te jeuken.

If you move another element to the front, like Door die nieuwe trui, the verb still stays in second position, so the subject moves after it:

• Door die nieuwe trui begonnen mijn armen te jeuken.

So this is not random word order. It is a very common Dutch pattern:

• Gisteren ging ik naar school.
• In de zomer werken we veel.
• Door de kou bibberde hij.

The verb beginnen is often followed by te + infinitive in Dutch.

So:

• beginnen te jeuken = to begin to itch
• beginnen te lachen = to begin to laugh
• beginnen te regenen = to begin to rain

In your sentence:

• mijn armen begonnen te jeuken = my arms started itching / began to itch

This is a very standard construction.

Jeuken means to itch.

In Dutch, just like in English, the body part doing the itching can be the subject:

• Mijn armen jeuken. = My arms itch.
• Mijn huid jeukt. = My skin itches.

So:

• mijn armen begonnen te jeuken literally means my arms began to itch

That is why mijn armen is the subject of the verb begonnen.

After verbs of perception such as zien (to see), horen (to hear), and sometimes voelen (to feel), Dutch normally uses a bare infinitive, without te.

So:

• Ik zag Tom zweten. = I saw Tom sweating.
• Ik hoorde haar zingen. = I heard her singing.
• We voelden de grond trillen. = We felt the ground shaking.

That is why te is not used here.

Compare:

• Tom begon te zweten. = Tom began to sweat
• Ik zag Tom zweten. = I saw Tom sweating

So beginnen takes te, but zien does not.

Because zag is the finite verb, and zweten is the infinitive that goes with it.

In Dutch main clauses, the finite verb usually comes early, while other verbs often appear later in the clause. So:

• ik zag Tom zweten
• literally: I saw Tom sweat/sweating

This word order is normal with perception verbs:

• Ik zag hem lopen.
• Wij hoorden de baby huilen.
• Ze voelde haar hart bonzen.

Here, ook means also / too.

In:

• ik zag Tom ook zweten

the most natural reading is that Tom was also sweating.

So the speaker is describing one thing happening to themselves, and then adds that Tom was experiencing something similar too.

Depending on context, ook can shift emphasis depending on placement, but here it most naturally goes with Tom zweten:

• I saw Tom sweating too / also

Volle is the inflected form of vol (full).

Because bus is a de-word and the adjective comes before the noun with a definite article (de), Dutch adds -e:

• de volle bus = the full/crowded bus

This is standard adjective inflection:

• de nieuwe trui
• de volle bus
• het nieuwe huis

In this sentence, de volle bus is best understood as the crowded bus rather than literally just the full bus.

Not in this sentence.

Door can mean different things, including:

• through
• by
• because of

Here it means because of:

• Door die nieuwe trui = Because of that new sweater

That use is very common when talking about a cause:

• Door de regen bleef ik thuis. = Because of the rain, I stayed home.
• Door het lawaai kon ik niet slapen. = Because of the noise, I couldn’t sleep.

So here door is expressing the reason the speaker’s arms started itching.

Because trui is a de-word:

• de trui = the sweater

For singular de-words, the demonstrative is die:

• die trui = that sweater

Compare:

• de trui → die trui
• het huis → dat huis

So:

• die nieuwe trui = that new sweater

Both are simple past forms.

• begonnen is the simple past plural of beginnen
• zag is the simple past singular of zien

Why plural for begonnen? Because the subject is mijn armen, which is plural.

Compare:

• Mijn arm begon te jeuken. = My arm began to itch.
• Mijn armen begonnen te jeuken. = My arms began to itch.

And:

• Ik zag Tom. = I saw Tom.

So the sentence is narrating past events in the simple past, which is very common in written Dutch and also natural in storytelling.

Yes, in many contexts that would be possible.

• Mijn armen begonnen te jeuken
• Mijn armen gingen jeuken

Both can mean my arms started itching.

However, beginnen te + infinitive is a very direct and standard way to express beginning to do something.
Gaan + infinitive can also express the start of something, but it is broader and can sometimes sound a bit different depending on context.

So in this sentence, begonnen te jeuken is very natural and precise.

They are used a little differently.

• jeuken describes a sensation or condition: to itch
• zweten describes a physical action/process: to sweat

In the sentence:

• mijn armen begonnen te jeuken = the speaker’s arms started itching
• ik zag Tom ook zweten = the speaker saw Tom sweating

So the first part focuses on what the speaker experienced physically, and the second part focuses on what the speaker observed happening to Tom.