Я взяла длинную верёвку, чтобы привязать коробку в багажнике.

Because in the past tense, Russian verbs agree with the subject’s gender in the singular.

• я взяла = a woman/girl is speaking
• я взял = a man/boy is speaking

So this sentence tells you the speaker is female. If a male speaker said it, it would be:

Я взял длинную верёвку, чтобы привязать коробку в багажнике.

Because it is the direct object of взяла: the speaker took the rope.

That means верёвка has to go into the accusative case:

• длинная верёвка = nominative, dictionary form
• длинную верёвку = accusative

The adjective must agree with the noun, so both words change:

• длинная → длинную
• верёвка → верёвку

This is a very common feminine singular pattern for nouns ending in -а and their adjectives.

Russian has no articles, so there is no direct equivalent of a or the.

So длинную верёвку can mean:

• a long rope
• the long rope

Context tells you which meaning is intended. If Russian wants to be more specific, it can use other words such as:

• эту длинную верёвку = this long rope
• одну длинную верёвку = one/a certain long rope

Here чтобы introduces a purpose clause. It means:

• to
• in order to
• so as to

So:

Я взяла длинную верёвку, чтобы привязать коробку...

means the rope was taken for the purpose of tying the box.

A useful pattern is:

• чтобы + infinitive when the subject is the same
• чтобы + past-tense-like verb form when the subject is different

Compare:

• Я взяла верёвку, чтобы привязать коробку.
  The same person took the rope and will tie the box.

• Я взяла верёвку, чтобы он привязал коробку.
  The speaker took the rope, but he will tie the box.

Because привязать is perfective, and here Russian focuses on the result: getting the box tied.

• привязать = to tie/fasten successfully, to get it done
• привязывать = to be tying, to tie repeatedly, or to talk about the process in a more general way

After чтобы, Russian often uses the perfective infinitive when the goal is a completed result.

So чтобы привязать коробку means something like in order to tie/secure the box.

They are doing different jobs, so they use different cases.

• коробку is the direct object of привязать, so it is in the accusative
• в багажнике shows location, so after в it takes the prepositional case

Breakdown:

• коробка → коробку
• багажник → в багажнике

So:

• привязать коробку = to tie the box
• в багажнике = in the trunk

Because в can take two different cases depending on meaning:

• в + accusative = motion into somewhere
• в + prepositional = location in somewhere

So:

• в багажник = into the trunk
• в багажнике = in the trunk

In your sentence, the meaning is location, not movement, so Russian uses в багажнике.

In normal usage, it will usually be understood as meaning that the box is in the trunk when the tying happens.

So the overall idea is something like:

• the speaker took a rope
• to tie down the box
• the box is in the trunk

Like in English, this kind of phrase can sometimes be a little flexible, but the natural reading is that the trunk is the location of the box/the tying situation.

If you wanted to make it more explicit, Russian could say something more detailed, for example:

• ...чтобы привязать коробку, которая была в багажнике.
• ...чтобы привязать коробку, лежавшую в багажнике.

Yes. Russian often omits subject pronouns when the subject is already clear from context.

So you could also say:

Взяла длинную верёвку, чтобы привязать коробку в багажнике.

That would still sound natural if it is already obvious who is speaking. Using я is also fine; it can make the sentence clearer or slightly more explicit.

верёвку is pronounced roughly vye-RYOF-ku.

A few important points:

• ё is pronounced like yo
• when ё is written, it is almost always stressed
• many Russian texts replace ё with е, so you may also see веревку, but the pronunciation is still верёвку

So this word is not pronounced as if it had a plain е in the stressed syllable.

Yes, верёвка is a very normal everyday word for rope.

Depending on context, Russian might also use other words:

• канат = a thick rope, cable, or hawser
• шнур = cord
• бечёвка = twine/string-like cord

So верёвка is a good neutral choice here for an ordinary rope used to tie something down.