Она попросила разрешения.

Because it is past tense feminine singular. In Russian, past-tense verbs agree with the subject in gender and number:

• он попросил
• она попросила
• оно попросило
• они попросили

Since the subject here is она, the verb must be попросила.

The dictionary form is попросить. It is a perfective verb, so it presents the action as a completed single request.

Its imperfective partner is просить:

• Она попросила разрешения = she made the request / asked once
• Она просила разрешения = she was asking, used to ask, or asked repeatedly

In English, both may simply be translated as asked, so this is an important Russian aspect difference to notice.

Russian does not use a separate word like for here. Instead, it uses a case pattern.

The phrase is:

• попросить разрешения

Here разрешения is in the genitive singular, and that helps express the idea of ask for permission.

This is a very common pattern with просить / попросить, especially in expressions like:

• просить помощи
• просить совета
• просить прощения
• просить разрешения

So it is best to learn попросить разрешения as a whole chunk.

Here it is singular: the genitive singular of разрешение.

The form разрешения can look plural if you see it by itself, but in this sentence it is singular because the expression попросить разрешения means to ask for permission as an abstract thing, not to ask for permissions.

Because after просить / попросить, the thing being requested is often put in the genitive, especially in fixed, common expressions with abstract nouns.

So Russian normally says:

• просить разрешения
• попросить разрешения

For a learner, the safest approach is to memorize this as the standard idiomatic form.

You can add у + genitive:

• Она попросила разрешения у начальника.
• Она попросила у начальника разрешения.

Both mean She asked the boss for permission.

Russian often leaves this out if the person is already clear from the context.

Yes. Спросить разрешения is also very common.

The difference is small:

• спросить focuses more on asking/inquiring
• попросить focuses a bit more on making a request

In many everyday situations, both are natural.

Yes. This is the neutral, standard word order.

Russian word order is flexible, so you can move things around for emphasis:

• Разрешения она попросила. = emphasis on what she asked for
• У начальника она попросила разрешения. = emphasis on who she asked

But Она попросила разрешения is the default way to say it.

Yes. Russian often adds an infinitive after разрешения:

• Она попросила разрешения уйти. = She asked permission to leave.
• Она попросила разрешения войти. = She asked permission to come in.

So Она попросила разрешения. is complete by itself, but it can also be the start of a longer sentence.

The stress is:

• она́
• попроси́ла
• разреше́ния

A rough pronunciation guide is:

• a-NA
• pa-pra-SEE-la
• raz-ra-SHE-ni-ya

Also remember that unstressed о is reduced in pronunciation, so it may sound less like a clear English o than the spelling suggests.