Learn Hardcore Russian: После ссоры лучше помириться сразу. - After a quarrel, it is better to make up right away.

Because после always requires the genitive case. The dictionary form is ссора (quarrel, argument), and its genitive singular form is ссоры. So после ссоры means after a quarrel or after an argument.

What does лучше mean here?

Лучше literally means better. In this sentence, it has the sense of it is better to... or you had better... So the sentence is giving general advice: after an argument, it is better to make up right away.

Why is there no word for it is?

In Russian, the verb to be is usually omitted in the present tense. So where English says It is better, Russian often just says Лучше. That is completely normal.

Why is помириться in the infinitive?

After words like лучше, Russian often uses the infinitive to express advice or recommendation:

лучше подождать = it’s better to wait
лучше уйти = it’s better to leave
лучше помириться = it’s better to make up

So помириться is in the infinitive because it names the action being recommended.