Она молча выслушала брата, и им удалось быстро уладить этот вопрос.

Выслушала comes from выслушать, which means to listen to someone to the end / hear someone out.

This is different from слушать, which simply means to listen.

So:

• слушала брата = she was listening to her brother
• выслушала брата = she listened to her brother fully, until he finished

The prefix вы- adds the idea of completion here, and выслушать is perfective, so it fits a completed action in the past.

Because the subject is она, which is feminine singular.

In the past tense, Russian verbs agree with the subject in gender and number:

• он выслушал
• она выслушала
• оно выслушало
• они выслушали

So выслушала tells you that the person who did the action is feminine.

Because молча is an adverb, not an adjective.

It means silently or without saying anything, and adverbs do not change for gender, number, or case.

Compare:

• молчаливая женщина = a silent woman
  
  • adjective, so it changes
• она молча слушала = she listened silently
  
  • adverb, so it does not change

Here молча describes how she listened.

Брат is the dictionary form, which is the nominative singular.

In this sentence, брата is in the accusative case because he is the direct object of выслушала: she listened to whom? — her brother.

For masculine animate nouns, the accusative singular looks like the genitive singular:

• брат = nominative
• брата = accusative/genitive singular

This is very common in Russian:

• Я вижу брата.
• Она выслушала брата.

Им удалось is a very common Russian construction meaning they managed to... or they succeeded in...

Literally, it is closer to:

• To them, it succeeded
• It worked out for them

That is why:

• им is in the dative
• удалось is in neuter singular past tense

Russian often uses this impersonal pattern where English would use a personal subject:

• Им удалось уладить вопрос. = They managed to settle the issue.
• Мне удалось найти книгу. = I managed to find the book.

Они смогли is also possible in some contexts, but it is a bit more direct: they were able to.
Им удалось often sounds more like they succeeded in managing it, sometimes with a slight sense that it was not easy.

Им is the dative plural form of они / they.

In this sentence, it refers to the two people involved: she and her brother.

So the meaning is:

• She listened to her brother silently,
• and the two of them managed to settle the issue quickly.

Russian often avoids repeating nouns or pronouns when the reference is clear from context.

Because удалось is part of an impersonal construction.

The verb is not agreeing with им. In fact, им is not the grammatical subject here; it is in the dative case.

The idea is:

• им удалось = it worked out for them / they managed

In this pattern, Russian uses neuter singular past tense:

• мне удалось
• тебе удалось
• ему удалось
• им удалось

So удалось stays the same regardless of who the dative experiencer is.

Because удаться / удалось is commonly followed by an infinitive to say manage to do something.

Pattern:

• кому? + удалось + infinitive

Examples:

• Мне удалось понять правило. = I managed to understand the rule.
• Им удалось быстро уладить этот вопрос. = They managed to settle this issue quickly.

So уладить names the action they succeeded in doing.

Уладить means to settle, to resolve, to smooth over, or to sort out, especially a problem, disagreement, or practical issue.

It is a perfective verb, so it presents the action as completed.

Common combinations include:

• уладить вопрос = settle an issue
• уладить конфликт = resolve a conflict
• уладить проблему = sort out a problem

In this sentence, этот вопрос is something they managed to resolve successfully.

Because вопрос is the direct object of уладить, so it is in the accusative singular.

For an inanimate masculine noun like вопрос, the accusative looks the same as the nominative:

• этот вопрос = nominative
• уладить этот вопрос = accusative

Compare that with an animate masculine noun like брат:

• брат → брата

That is why you see:

• выслушала брата
• уладить этот вопрос

Быстро is an adverb meaning quickly, and here it modifies the action уладить.

Russian word order is fairly flexible, so им удалось быстро уладить этот вопрос is the natural, neutral way to say:

• they managed to settle this issue quickly

You could move words around for emphasis, but this version sounds standard and smooth.

For example:

• Им удалось быстро уладить этот вопрос. = neutral
• Им удалось уладить этот вопрос быстро. = more emphasis on quickly

Because the sentence presents a chain of completed results:

• выслушала = she listened to him all the way through
• удалось = they succeeded
• уладить = settle/resolve completely

This is very typical in Russian narration when the speaker is talking about finished events.

If imperfective verbs were used instead, the meaning would shift toward process, repetition, or incompleteness. Here the point is that one action was completed, and then as a result they resolved the matter.

The comma separates two clauses:

1. Она молча выслушала брата
2. им удалось быстро уладить этот вопрос

They are connected by и = and, but each clause has its own predicate:

• выслушала
• удалось

So Russian punctuation normally requires a comma here.