В поезде ей было прохладно, поэтому она надела тёплую жилетку поверх блузки.

In Russian, в поезде literally means in the train, and that is the normal way to say on the train when you mean being inside it as a passenger.

A useful pattern is:

• в
  • location you are physically inside
• на
  • surface, event, or some forms of transport in a more abstract sense

So Russian says:

• в поезде = on the train
• в автобусе = on the bus
• в машине = in the car
• в метро = on the metro / in the subway

But:

• на самолёте can mean by plane
• на поезде often means by train rather than inside the train

So in this sentence, в поезде is about location, not means of travel.

This is a very common Russian pattern for talking about how someone feels physically or emotionally.

Russian often uses an impersonal construction:

• мне холодно = I am cold
• ему жарко = he is hot
• ей было прохладно = she felt a bit cool / she was a little chilly

Literally, ей было прохладно is something like:

• to her it was cool

The person experiencing the feeling goes in the dative case:

• мне = to me
• тебе = to you
• ему / ей = to him / her
• нам = to us

This is one of the most natural ways in Russian to describe temperature-related feelings.

Because было is used in this kind of impersonal sentence.

In ей было прохладно, there is no real grammatical subject like она doing the action. The idea is just that a state existed:

• было прохладно = it was cool

In Russian, impersonal past-tense constructions usually use neuter singular past tense:

• было холодно
• было жарко
• было грустно
• было приятно

So было does not agree with ей. The word ей is only the experiencer in the dative, not the subject.

Прохладно means cool, a bit chilly, or not very warm. It is milder than холодно.

So:

• холодно = cold
• прохладно = cool / somewhat chilly

In this sentence, прохладно suggests that she was uncomfortable enough to put on another layer, but it was probably not freezing or severely cold.

Надела is the past tense of надеть, which means to put on a piece of clothing.

So:

• надеть куртку = to put on a jacket
• надеть шапку = to put on a hat
• она надела жилетку = she put on a vest

The ending -а in надела shows that the subject is feminine:

• он надел = he put on
• она надела = she put on

Also, надеть is perfective, so надела presents the action as completed: she put it on.

This is a classic Russian learner question.

The standard distinction is:

• надеть = to put something on oneself or onto someone
• одеть = to dress someone

Examples:

• надеть пальто = to put on a coat
• надеть ребёнку шапку = to put a hat on the child
• одеть ребёнка = to dress the child

So in this sentence, она надела тёплую жилетку is correct because the clothing item itself is the object.

A common memory trick:

• надеть что? = put on what?
• одеть кого? = dress whom?

Because жилетку is the direct object of надела, so it goes into the accusative case.

The dictionary forms are:

• тёплая = warm
• жилетка = vest

But after надела, we need the accusative feminine singular:

• тёплую жилетку

The adjective must agree with the noun in:

• gender
• number
• case

So:

• nominative: тёплая жилетка
• accusative: тёплую жилетку

This is normal adjective-noun agreement.

Yes, a little.

Жилетка means vest or waistcoat, and it often sounds a bit more everyday and practical than жилет. In many contexts, жилетка is what you would naturally call a warm sleeveless layer of clothing.

So тёплая жилетка sounds very natural for an extra warm layer worn over a blouse.

Because поверх is used here as a preposition meaning over or on top of, and it normally takes the genitive case.

So:

• поверх блузки = over the blouse
• поверх рубашки = over the shirt
• поверх платья = over the dress

The dictionary form is блузка, but after поверх it becomes блузки.

This phrase tells you the order of the clothes: first the blouse, then the vest on top.

Because поэтому here means therefore / so, linking two clauses:

• В поезде ей было прохладно
• поэтому она надела тёплую жилетку поверх блузки

Russian normally uses a comma before поэтому when it connects one clause to the result in the next clause.

So the structure is:

• cause: it was cool for her on the train
• result: so she put on a warm vest

The sentence has very natural, neutral word order.

• В поезде sets the scene first: where this happened
• ей было прохладно gives her physical state
• поэтому introduces the result
• она надела тёплую жилетку поверх блузки gives the action

Russian word order is flexible, but this version flows naturally:

1. location
2. feeling
3. consequence
4. action

You could change the order for emphasis, but the original sounds normal and clear.