Горячий бульон помог ей согреться после дождя.

Because it agrees with бульон.

• бульон is masculine, singular, nominative
• so the adjective must also be masculine, singular, nominative

That gives:

• горячий бульон = hot broth

If the noun changed, the adjective would change too:

• горячая вода = hot water
• горячее молоко = hot milk
• горячие супы = hot soups

Бульон is in the nominative case because it is the subject of the sentence — the thing doing the action of helping.

So in:

• Горячий бульон помог ей согреться после дождя.

the broth is what helped, so бульон stays nominative.

Because the verb помочь / помогать takes the person being helped in the dative case.

• кто? горячий бульон
• помог кому? ей

So:

• она = she
• ей = to her

This is very normal Russian grammar:

• Я помог брату. = I helped my brother.
• Она помогла мне. = She helped me.
• Мы помогли ей. = We helped her.

In the past tense, Russian verbs agree with the gender and number of the subject.

The subject here is бульон, which is:

• masculine
• singular

So the past tense form is:

• помог = masculine singular

Compare:

• Сестра помогла. = Sister helped.
• Письмо помогло. = The letter helped.
• Друзья помогли. = Friends helped.

Согреться means to get warm / to warm oneself up.

It is:

• an infinitive
• perfective
• reflexive

After помочь, Russian often uses an infinitive:

• помочь сделать что-то = to help do something

So:

• помог ей согреться = helped her get warm

The perfective form согреться suggests reaching the result: she became warm.

This is a very common question.

• согреть = to warm something/someone
• согреться = to get warm / warm oneself

Examples:

• Она согрела чай. = She warmed the tea.
• Она согрелась чаем. = She got warm with tea.

In your sentence, the woman is the one who becomes warm, so Russian uses the reflexive form:

• согреться

not

• согреть

The ending -ся makes the verb reflexive.

Very roughly, it shows that the action relates back to the subject. Here the meaning is not literally she warmed herself with her own hands, but rather she became warm.

So:

• греть = to warm
• согреть = to warm something up
• согреться = to warm up / get warm

In English, this is often translated naturally as get warm rather than warm oneself.

Because the preposition после always takes the genitive case.

So:

• дождь = rain
• после дождя = after the rain

Other examples:

• после урока = after class
• после работы = after work
• после фильма = after the film

So the form дождя is just the genitive singular of дождь.

It can often suggest the broader situation, not just a strict weather event.

In this sentence, после дождя naturally means something like:

• after the rain
• after she got wet in the rain
• after coming in from the rain

Russian often leaves that kind of context unstated if it is obvious.

Because Russian infinitives do not use a separate word like English to.

The infinitive is simply the verb form itself:

• согреться = to get warm
• читать = to read
• идти = to go

So:

• помог ей согреться literally = helped her get warm

That structure is completely normal in Russian.

Yes, and the difference is about aspect.

• помог = perfective → a completed helping action, seen as a whole
• помогал = imperfective → was helping / used to help / helped in an ongoing sense

In your sentence:

• Горячий бульон помог ей согреться

means the broth actually had the result of helping her warm up.

If you said:

• Горячий бульон помогал ей согреться

it would sound more like the hot broth was helping her get warm, focusing on process rather than completed result.

Yes. Russian word order is flexible, because case endings show the grammatical roles.

The neutral order here is:

• Горячий бульон помог ей согреться после дождя.

But you could also say:

• После дождя горячий бульон помог ей согреться.
• Ей помог согреться горячий бульон после дождя.

The meaning stays basically the same, but the emphasis changes.

For example:

• После дождя at the start emphasizes the situation first.
• Ей near the start can emphasize her.

A native speaker would understand it very naturally as something like:

• The hot broth helped her warm up after the rain.
• The hot broth helped her get warm after being out in the rain.

It sounds normal and idiomatic. The key ideas are:

• горячий бульон = the thing that helped
• ей = the person affected
• согреться = become warm
• после дождя = after the rain / after coming in from the rain

The stress pattern is:

• горя́чий
• бульо́н
• помо́г
• ей
• согре́ться
• по́сле
• дождя́

So a stress-marked version is:

• Горя́чий бульо́н помо́г ей согре́ться по́сле дождя́.

A few pronunciation notes:

• final г in помог sounds like к at the end: roughly памок
• unstressed vowels are reduced in normal speech
• ей is usually pronounced like yay or yey, depending on accent and phonetic description

Usually бульон is more specifically broth rather than soup.

• бульон = broth, clear stock
• суп = soup

So горячий бульон is best translated as hot broth.

That said, in some contexts learners may loosely think of it as a kind of light soup, but broth is the more accurate word here.