Мне нужно объясниться с братом после ссоры.

Because Russian often uses the dative case with words like нужно, надо, можно, and нельзя.

So мне нужно literally means something like:

To me, it is necessary...

That is why the person who feels the need is in the dative:

• я = I
• мне = to me / for me

So:

• Я должен объясниться... = I must explain myself...
• Мне нужно объясниться... = I need to explain myself / clear things up...

Both are possible, but the structure is different.

Нужно is an impersonal predicate meaning necessary / need to.

In this pattern, Russian does not use a personal verb like English need. Instead, it uses:

[dative person] + нужно + infinitive

So:

• Мне нужно объясниться = I need to explain myself / clear things up
• Тебе нужно идти = You need to go
• Нам нужно поговорить = We need to talk

A useful comparison:

• мне нужно = I need to
• я должен = I must / I am supposed to

Нужно often sounds a bit more neutral and less forceful than должен.

Объясниться does not simply mean to explain in the basic sense.

Here it means something more like:

• to explain oneself
• to clear things up
• to have an honest talk
• to sort things out verbally

After ссора (a quarrel), объясниться usually suggests that the speaker wants to talk through the conflict with the other person.

Compare:

• объяснить = to explain something
• объясниться = to explain oneself / clear things up with someone

So this is not just giving information. It is more personal and relational.

The -ся ending often makes a verb reflexive, but reflexive verbs in Russian do not always translate literally as oneself.

In this case, объясниться is a set verb with its own meaning. It means:

• to explain oneself
• to clear the air
• to sort things out by talking

So the -ся is part of the verb’s normal dictionary form here.

This is different from:

• объяснить что-то кому-то = to explain something to someone
• объясниться с кем-то = to have an explanation / clear things up with someone

So yes, the -ся matters a lot: it changes the construction and the meaning.

Because объясниться is used with с + instrumental when it means to clear things up with someone.

So:

• с братом = with my brother
• братом is the instrumental singular of брат

This construction is normal:

• объясниться с другом
• объясниться с родителями
• объясниться с начальником

If you used объяснить, the grammar would be different:

• объяснить брату ситуацию = to explain the situation to one’s brother

So:

• объясниться с братом = clear things up with my brother
• объяснить брату = explain something to my brother

They are related, but not the same structure.

Because the preposition после always takes the genitive case.

The noun is:

• ссора = quarrel, argument

Its genitive singular form is:

• ссоры

So:

• после ссоры = after the quarrel / after an argument

Other examples:

• после работы = after work
• после урока = after the lesson
• после разговора = after the conversation

So the important rule is:

после + genitive

Объясниться is perfective.

That makes sense here, because the speaker is referring to a single completed action: one talk, one attempt to clear things up.

The imperfective partner is объясняться.

Compare:

• Мне нужно объясниться с братом = I need to have that talk / get it over with
• Мне приходится объясняться с братом снова и снова = I have to keep explaining myself to my brother again and again

So in your sentence, the perfective is natural because the focus is on a specific, complete act after the quarrel.

Yes, but it would mean something different.

Мне нужно объяснить брату... means:

I need to explain something to my brother...

That structure usually expects an object, such as:

• Мне нужно объяснить брату ситуацию = I need to explain the situation to my brother
• Мне нужно объяснить брату, что произошло = I need to explain to my brother what happened

But Мне нужно объясниться с братом means:

I need to clear things up with my brother

So:

• объяснить брату = explain something to him
• объясниться с братом = talk things out with him

Sometimes yes, but the nuance is different.

• помириться = to make up / reconcile
• объясниться = to clear things up / explain oneself / talk it out

So:

• Мне нужно помириться с братом после ссоры means the focus is on restoring the relationship.
• Мне нужно объясниться с братом после ссоры means the focus is on having the conversation and sorting out what happened.

Often both ideas can go together:

First you объясняетесь, and then you миритесь.

Yes. Russian word order is flexible, and the cases show the grammatical relationships.

The original sentence is a neutral, natural order:

Мне нужно объясниться с братом после ссоры.

You could also say:

• После ссоры мне нужно объясниться с братом.
  This emphasizes after the quarrel.

• С братом мне нужно объясниться после ссоры.
  This can emphasize with my brother rather than with someone else.

The basic meaning stays the same, but the emphasis changes. The original version is a very normal default sentence.