Mia fratino portas varman koltukon, ĉar sen ĝi ŝia gorĝo fariĝas malvarma.

Because the possessive mia already makes the noun definite. In normal Esperanto, mia fratino naturally means my sister.

Using la with a possessive is usually unnecessary and is mostly used for special emphasis or contrast. So mia fratino is the standard, natural form here.

It can mean either, depending on context. The basic idea of porti is to carry / to bear / to have on oneself.

With clothing, it very often means to wear. So in this sentence, portas varman koltukon means is wearing a warm scarf.

Because koltukon is the direct object of portas, so it takes the accusative ending -n.

Adjectives in Esperanto must agree with the nouns they describe in:

• number
• case

So if the noun is koltukon, the adjective must match it: varman koltukon.

Because malvarma is not a direct object here. It is a predicate adjective after fariĝas, describing the subject ŝia gorĝo.

In other words:

• koltukon = object, so it gets -n
• malvarma = description of gorĝo, so it stays in the basic adjective form

This is similar to how English says her throat becomes cold, not becomes coldly and not with any object marking.

This is a very common learner question. Sia is a reflexive possessive, and it refers to the subject of its own clause.

In the clause sen ĝi ŝia gorĝo fariĝas malvarma, the grammatical subject is ŝia gorĝo. Since fratino is not the subject of that clause, sia would not be correct there.

So Esperanto uses ŝia to mean her throat = the throat of the sister.

Because sen is a preposition, and prepositions normally take the basic noun or pronoun form, not the accusative.

So:

• ĝi = it
• sen ĝi = without it

You do not add -n here, because ĝi is not a direct object.

It refers to the scarf, that is, koltuko.

Esperanto commonly uses ĝi for things and non-person referents. Since a scarf is not a person, ĝi is the natural pronoun to use.

Ĉar introduces a full reason clause, so it works perfectly here:

ĉar sen ĝi ŝia gorĝo fariĝas malvarma

That whole part is a complete clause with its own subject and verb. Pro is usually used before a noun phrase, not a full clause.

So:

• ĉar = because
• pro la malvarmo = because of the cold

If you wanted to use pro with a clause, you would usually need something heavier like pro tio, ke.... In this sentence, ĉar is simpler and more natural.

Fariĝi means to become or to get. So ŝia gorĝo fariĝas malvarma means her throat becomes cold or her throat gets cold.

The ending -as shows present tense, so fariĝas means becomes / gets.

Yes, in many contexts iĝas can also mean becomes. So ŝia gorĝo iĝas malvarma would be understandable.

However, fariĝas is very common and often feels a bit fuller or more standard in sentences like this. For a learner, fariĝas malvarma is an excellent choice.

Yes. Esperanto word order is fairly flexible because the endings show the grammatical roles.

For example, you could also say:

Mia fratino portas varman koltukon, ĉar ŝia gorĝo fariĝas malvarma sen ĝi.

That has basically the same meaning. The original order puts sen ĝi earlier, which emphasizes the idea of without it.

Gorĝo means throat, while kolo means neck.

A learner might expect a scarf to protect the kolo, and that is true. But saying the throat gets cold is also natural, because the cold affects that area and can make someone feel ill. So ŝia gorĝo fariĝas malvarma is perfectly understandable.