Я сожалею, что ответил слишком резко, — сказал я, и она виновато улыбнулась.

Here что means that, not what. It introduces a subordinate clause explaining what the speaker regrets:

Я сожалею, что ответил слишком резко
= I regret that I answered too sharply

So что ответил слишком резко is the content of the regret.

Сожалеть means to regret. Я сожалею is a fairly neutral or slightly formal way to express regret.

It is different from:

• Извините / прости = sorry / forgive me
• Мне жаль = I’m sorry / I feel bad
• Я жалею usually means I pity or I feel sorry for, so it is not the best choice here

So Я сожалею focuses on regret about one’s own action.

Because the two verbs refer to different times.

• сожалею = I regret now, in the present
• ответил = answered earlier, in the past

Russian subordinate clauses use the tense that fits the action itself, just like English:

I regret that I answered too sharply.

So there is nothing unusual here.

This is about verbal aspect.

• ответил is perfective
• отвечал is imperfective

Ответил is used because the speaker is talking about one completed reply: he answered once, and that answer was too sharp.

If you used отвечал, it would sound more like a process, repetition, or background action, which does not fit as well here.

So что ответил слишком резко means that I gave a sharp reply as a completed event.

Russian often omits subject pronouns when they are clear from context.

In Я сожалею, что ответил слишком резко, the subject of ответил is understood to be the same speaker, я.

Also, the past-tense form ответил shows masculine singular, which helps identify the speaker as male. If the speaker were female, it would be ответила.

So the full idea is:

Я сожалею, что я ответил слишком резко

but the second я is normally left out because it is unnecessary.

Слишком means too or overly.

Резко is an adverb meaning sharply, harshly, abruptly, or bluntly, depending on context.

So:

ответил слишком резко
= answered too sharply / too harshly / too abruptly

Here резко describes how he answered.

Because it modifies a verb, not a noun.

• резкий = adjective, used with nouns
• резко = adverb, used with verbs

Here the word describes the action ответил:

How did he answer?
Резко

So Russian needs the adverb резко.

This is standard Russian punctuation for direct speech followed by author words.

The spoken words come first:

Я сожалею, что ответил слишком резко

Then comes the reporting clause:

— сказал я

Because the spoken sentence would naturally end with a period, Russian changes that to a comma before the dash when author words follow.

Then the narration continues:

и она виновато улыбнулась

There is no second dash because the direct speech does not continue after сказал я.

Both are possible, but сказал я is very common in narration after direct speech.

Russian often uses verb-first order in reporting clauses:

• — сказал я
• — спросил он
• — ответила она

This sounds natural and literary.
Я сказал is also grammatical, but after direct speech сказал я is especially common.

Виновато means guiltily, with a guilty look, or sometimes apologetically.

So:

она виновато улыбнулась
= she smiled guiltily
or
= she gave an apologetic smile

Like резко, виновато is an adverb. It describes how she smiled.

It does not mean that she literally said she was guilty in a legal sense. It describes her expression or emotional tone.

Улыбнулась is past tense, feminine singular, perfective.

It tells us two things:

1. The subject is feminine: она
2. The smile is treated as one completed action

So она виновато улыбнулась means she gave a single guilty smile.

If the subject were masculine, it would be улыбнулся.

Yes, that is the natural reading.

The conjunction и simply means and, but in context it links her reaction to his words. Russian often leaves that relationship to context rather than spelling it out.

So the sentence suggests:

• he said he regretted answering harshly
• she reacted with a guilty or apologetic smile

The connection is understood even though Russian does not explicitly say in response or because of that.