У моей жены ручная кладь была лёгкой, поэтому она взяла её с собой в салон.

Literally, у моей жены means at my wife or by my wife, but in Russian this structure is very commonly used to express possession:

У моей жены ручная кладь была лёгкой
= My wife’s hand luggage was light
or more literally
= My wife had hand luggage that was light

Russian often prefers у + genitive for someone has something rather than using a verb like to have.

So:

• У меня есть книга = I have a book
• literally: At me there is a book

Using иметь is possible in Russian, but it is much less common in everyday speech and often sounds more formal, abstract, or bookish.

Because у requires the genitive case.

The base form is:

• моя жена = my wife

After у, both words must change to genitive:

• у моей жены

Breakdown:

• жена → жены
• моя → моей

This is just normal adjective-pronoun agreement: the possessive word моя has to match the noun жена in case, number, and gender.

Ручная кладь means hand luggage, carry-on baggage, or carry-on.

It is a fixed travel expression:

• ручная = hand-, carried by hand
• кладь = load, baggage in this expression

So the whole phrase means luggage you take with you into the aircraft cabin, not checked baggage.

This is the normal Russian term you will see at airports and in airline rules.

Because the main noun is кладь, and кладь is feminine.

That is why the adjective agrees with it:

• ручная кладь
• лёгкая кладь in nominative
• лёгкой when the grammar requires instrumental

A common thing for learners to remember is that nouns ending in ь can be either masculine or feminine, so you have to learn the gender individually. Кладь happens to be feminine.

Here лёгкой is in the instrumental case, and Russian often uses instrumental after быть in the past or future when describing what something was like.

So:

• ручная кладь была лёгкой

This is a normal and natural structure.

You may also hear:

• ручная кладь была лёгкая

That is also possible. The difference is mostly stylistic rather than a big change in meaning.

Very roughly:

• лёгкой can sound a bit more neutral, standard, or literary
• лёгкая can sound a bit more direct or conversational

For a learner, the important point is: after был / была / было / были, Russian may use either nominative or instrumental with predicate words, and лёгкой here is perfectly normal.

Поэтому means therefore, so, or that’s why. It introduces a result.

So the sentence means:

• the hand luggage was light
• therefore / so she took it into the cabin

By contrast, потому что means because and introduces a reason.

Compare:

• Ручная кладь была лёгкой, поэтому она взяла её с собой в салон.
  The luggage was light, so she took it into the cabin.

• Она взяла ручную кладь с собой в салон, потому что она была лёгкой.
  She took the hand luggage into the cabin because it was light.

Same basic idea, but the logical direction is different.

Взяла is the perfective past tense of взять, meaning took in the sense of a completed action.

Here it fits because the sentence describes one finished event:

• she took it with her into the cabin

If you used брала from брать instead, that would be imperfective, which usually suggests:

• repeated action
• process
• background action
• habitual behavior

So:

• она взяла её = she took it once, as a completed act
• она брала её = she was taking it / used to take it / would take it

In this sentence, the completed one-time action is what we want, so взяла is the natural choice.

Because её refers to ручную кладь, which is a feminine noun.

In Russian, things also have grammatical gender, and pronouns agree with that gender. So a feminine thing can be referred to by:

• она = it / she, depending on context
• её = it / her, depending on context

Here the object is the luggage:

• она взяла её = she took it

So even though English uses it for luggage, Russian uses the feminine pronoun because кладь is feminine.

С собой means with oneself / with her / along with her.

So:

• взяла её с собой в салон = took it with her into the cabin

It emphasizes that she did not check it in or leave it somewhere else; she personally carried it along.

Could you say just взяла её в салон? Yes, that is possible, but с собой is very natural and makes the meaning more explicit: she took it with her.

This phrase is extremely common:

• возьми с собой паспорт = take your passport with you
• я взял с собой воду = I took water with me

Because this sentence describes movement into the cabin.

Russian uses:

• в + accusative for motion into
• в + prepositional for location in

So:

• в салон = into the cabin
• в салоне = in the cabin

Compare:

• Она взяла её в салон. = She took it into the cabin.
• Она оставила её в салоне. = She left it in the cabin.

Yes, салон can mean salon in some contexts, but here it means the cabin or passenger compartment of a vehicle, especially an airplane.

In travel Russian, в салон often means:

• into the aircraft cabin
• into the bus interior
• into the car interior

So in this sentence, в салон is best understood as into the cabin.

Yes, Russian could sometimes omit the subject pronoun, especially when it is obvious from context:

• ...поэтому взяла её с собой в салон

That is possible.

But она is very helpful here because the previous part of the sentence contains another feminine noun phrase, ручная кладь, so adding она makes it immediately clear that the wife is the one doing the action, not the luggage.

So the pronoun improves clarity.

Yes, that alternative is also correct:

• Ручная кладь моей жены была лёгкой...

The version you were given:

• У моей жены ручная кладь была лёгкой...

puts early focus on my wife as the possessor. It sounds natural in context, especially if the conversation is about different people and their luggage.

Russian word order is more flexible than English word order, and changes in order often affect emphasis more than basic meaning.

So both are grammatical, but they do not highlight exactly the same thing.