После её слов я молча кивнул, хотя сначала хотел возразить.

Because после takes the genitive case, and слов is the genitive plural of слово.

• слово = word
• слова can mean either:
  • nominative plural: words
  • genitive singular: of the word
• слов = genitive plural: of words

So после её слов literally means after her words or more naturally after what she said.

Её means her.

In this sentence, it shows possession: её слов = her words / what she said.

A useful comparison:

• она = she
• её = her
• ей = to her
• ею / ей = by her

Here, only the possessive meaning is needed.

Also, you may see ее written without the dots. That is very common in normal writing. её is just the more precise spelling.

Literally, yes: after her words.

But in natural English, it usually means:

• after she said that
• after what she said
• following her words

Russian often uses слова in places where English would prefer what someone said.

So the phrase is grammatically literal, but idiomatically broader.

Молча is an adverb meaning silently, without saying anything, or in silence.

It comes from the idea of молчать = to be silent / to keep quiet.

So:

• я молча кивнул = I nodded silently
• more naturally: I nodded without saying anything

It emphasizes that the speaker did not respond verbally.

Кивнул is the perfective past form of кивнуть, meaning to nod once or to give a nod.

That fits this sentence well, because the action is a single completed event:

• кивнул = nodded once / gave a nod
• кивал = was nodding / nodded repeatedly / used to nod

Here the speaker did one completed action after hearing her words, so кивнул is the natural choice.

That is the past tense masculine singular ending.

Russian past tense agrees with gender and number:

• кивнул = he nodded
• кивнула = she nodded
• кивнуло = it nodded
• кивнули = they nodded

Since я can be said by a man or a woman, the form tells you the speaker is male here.

Because the whole sentence is in the past.

• хотел = wanted
• возразить = to object / to protest / to argue back

So сначала хотел возразить means at first I wanted to object.

This is a very common Russian pattern:

• хотел + infinitive = wanted to do something
• начал + infinitive = began to do something
• решил + infinitive = decided to do something

Because возразить is perfective, and here it means a single act of objection: to object, to speak up against something once.

Compare:

• возразить = to object once, to make an objection
• возражать = to be objecting, to object repeatedly, to have the habit of objecting

After хотел, Russian often chooses the aspect based on the intended action:

• хотел возразить = wanted to make an objection
• хотел возражать would sound more like wanting to engage in an ongoing process of objecting

So возразить is the better fit here.

Хотя means although, though, or even though.

It introduces a contrast:

• я молча кивнул = I nodded silently
• хотя сначала хотел возразить = although at first I wanted to object

So the idea is: the speaker ended up doing one thing, even though his initial impulse was different.

Сначала means at first or initially.

Here it shows that the speaker’s first reaction was to object, but then he changed course.

So:

• сначала хотел возразить = at first I wanted to object

It does not mean from the very beginning in a big historical sense here; it just means the initial reaction in that moment.

Because хотя introduces a subordinate clause.

Russian normally separates this kind of clause with a comma:

• После её слов я молча кивнул, хотя сначала хотел возразить.

This is similar to English punctuation before although in many sentences.

Yes, Russian word order is flexible, but the current order sounds very natural.

The sentence begins with После её слов to set the context first:

• После её слов = after what she said
• я = I
• молча кивнул = silently nodded

This ordering gives a smooth flow:

1. situation/context
2. subject
3. manner + action

Other orders are possible, for example:

• Я молча кивнул после её слов...
• Я после её слов молча кивнул...

But the original version sounds especially natural in narrative writing.

Yes, very common.

Russian often combines an adverb like молча with a short completed action:

• молча кивнул = nodded silently
• молча сел = sat down without saying anything
• молча ушёл = left without a word

It is a compact and natural way to describe action without speech.