Мы начали переговоры не потому, что был конфликт, а потому, что хотели лучше понять друг друга.

Начали is the past tense of the perfective verb начать. Perfective is used here because the sentence focuses on the fact that the negotiations started.

If you used начинали from начинать instead, it would usually sound more like:

• the action was in progress,
• it happened repeatedly,
• or it is being described as background rather than as a completed starting point.

So Мы начали переговоры = We began negotiations is the natural choice here.

Because the subject is мы = we.

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

• gender if singular
• number if plural

So:

• я начал / начала
• он начал
• она начала
• мы начали

That is why you see начали.

It is in the accusative case, because начать takes a direct object:
начать что? → переговоры

But for inanimate plural nouns, the accusative looks the same as the nominative, so переговоры does not change form here.

Also, переговоры meaning negotiations is normally used in the plural in Russian. So начали переговоры is the standard way to say started negotiations.

This is a very common Russian contrastive pattern. It means:

• not because ..., but because ...

So the speaker is correcting a possible assumption:

• not because there was a conflict
• but because we wanted to understand each other better

It is a neat way to reject one reason and give the real one.

The commas mark the structure of the sentence.

Two things are happening:

1. что introduces a subordinate clause:

   • потому, что был конфликт
   • потому, что хотели лучше понять...

2. The two reasons are contrasted with не ..., а ...

   • не потому ... , а потому ...

So the commas help show:

• the first reason clause,
• the contrast,
• and the second reason clause.

In this contrastive pattern, Russian often splits потому что into потому, что.

Because in the clause что был конфликт, the subject is конфликт, not мы.

• конфликт is singular masculine
• so the past-tense verb is был

Even though the whole sentence starts with мы, that мы does not control the verb inside this clause.

Russian has no articles, so there is no separate word for a or the.

Whether English would use a conflict or the conflict depends on context, not on a Russian article. In this sentence, был конфликт simply means something like there was a conflict.

Because Russian often drops the subject pronoun when it is already clear from the verb form or the context.

Here, хотели already tells you the subject is we:

• хотели = wanted in the plural form

So Russian does not need to repeat мы, although it could be added for emphasis:

• ...а потому, что мы хотели...

Without emphasis, leaving it out sounds natural.

After хотеть, Russian normally uses an infinitive to say what someone wants to do.

So:

• хотели понять = wanted to understand
• хотели узнать = wanted to find out
• хотели поговорить = wanted to talk

This is very similar to English want + to + verb.

Because понять is perfective and focuses on reaching understanding as a result.

• понимать = to understand, to be understanding, ongoing process/state
• понять = to come to understand, to grasp, to understand successfully

In this sentence, the idea is not just ongoing understanding, but achieving better mutual understanding. That is why понять fits well.

Лучше means better. It is the comparative form of хорошо.

So лучше понять means:

• to understand better
• to understand more fully
• to understand each other better than before

Russian often uses лучше exactly the way English uses better, even when the comparison is only implied.

Друг друга means each other.

It is a special Russian reciprocal expression, and it changes by case:

• accusative/genitive: друг друга
• dative: друг другу
• instrumental: друг другом
• prepositional: друг друге

Here it follows понять, which takes the accusative, so the correct form is друг друга.

So:

• понять друг друга = to understand each other

This is one of those forms that learners usually just have to get used to as a set expression.