El pan salió tan bien que mi madre cortó otra rebanada para acompañar la sopa.

Yes, salir often means to go out / to leave, but in Spanish it also has a very common extended meaning: to turn out, especially with food, plans, or results.

So:

• El pan salió bien = The bread turned out well
• La receta salió mal = The recipe turned out badly
• Todo salió perfecto = Everything turned out perfectly

This is a very natural use in everyday Spanish.

Because the sentence is talking about the result of making the bread, not just describing the bread.

• salió bien focuses on how it turned out in the end
• estaba bueno/bien would describe how it was
• fue bueno would usually mean it was good in a broader sense, but it is less natural here

With food, Spanish often uses salir bien/mal when talking about the outcome of cooking or baking.

Example:

• La tarta salió muy bien = The cake turned out very well

Tan... que means so... that.

In this sentence:

• tan bien = so well
• que = that

So:

• El pan salió tan bien que... = The bread turned out so well that...

This structure is very common:

• Era tan caro que no lo compré = It was so expensive that I didn’t buy it
• Hablaba tan rápido que no entendí nada = He/She spoke so fast that I didn’t understand anything

Because tan is used in the structure so... that.

Compare:

• muy bien = very well
• tan bien que... = so well that...

So:

• El pan salió muy bien = The bread turned out very well
• El pan salió tan bien que mi madre cortó otra rebanada = The bread turned out so well that my mother cut another slice

If there is a consequence introduced by que, tan is the natural choice.

Cortó is the preterite (simple past) of cortar for él/ella/usted.

Here it refers to mi madre, so:

• mi madre cortó = my mother cut

The preterite is used because this is a completed action in the past: she cut another slice at that moment.

Conjugation of cortar in the preterite:

• yo corté
• tú cortaste
• él/ella/usted cortó
• nosotros cortamos
• vosotros cortasteis
• ellos/ellas/ustedes cortaron

In Spanish, family words usually appear without an article when they are used with a possessive adjective.

So you say:

• mi madre
• tu hermano
• su abuela

Not:

• la mi madre ✘

Spanish normally uses:

• mi madre = my mother

You may sometimes see an article with family nouns in other structures, but not in this basic possessive pattern.

Otra rebanada means another slice.

• una rebanada = a slice
• otra rebanada = another slice

Using otra shows that there was already at least one slice before, and then she cut one more.

Also, rebanada is a common word for a slice, especially of bread, cake, etc.

Examples:

• una rebanada de pan = a slice of bread
• otra rebanada = another slice

Because rebanada is a feminine noun.

So the adjective must agree with it:

• otra rebanada ✔
• otro rebanada ✘

Agreement in Spanish:

• masculine singular: otro
• feminine singular: otra
• masculine plural: otros
• feminine plural: otras

Because the context already makes it clear that the slice is a slice of the bread just mentioned.

The sentence already starts with El pan..., so Spanish can naturally say just otra rebanada.

You could say:

• cortó otra rebanada de pan

but it is not necessary here because it would be slightly repetitive.

Para + infinitive means in order to / to when expressing purpose.

So:

• para acompañar la sopa = to go with the soup / in order to accompany the soup

This tells us why she cut another slice.

Examples:

• Compré pan para hacer bocadillos = I bought bread to make sandwiches
• Se sentó para descansar = He/She sat down to rest

Yes, literally it means to accompany the soup, but in natural English we would usually say:

• to go with the soup
• to have with the soup

Spanish often uses acompañar where English prefers go with.

Examples:

• Un vino blanco para acompañar el pescado = A white wine to go with the fish
• Pan para acompañar la sopa = Bread to go with the soup

Not naturally, if you want this exact meaning.

The sentence describes specific completed events:

• the bread turned out well
• the mother cut another slice

That is why the preterite is used:

• salió
• cortó

If you used the imperfect, it would suggest background, habit, or an ongoing situation, which does not fit as well here.

For example:

• El pan salía bien might suggest the bread was turning out well / used to turn out well
• mi madre cortaba otra rebanada might suggest my mother was cutting another slice / used to cut another slice

That changes the meaning.

No. Bien is an adverb here.

It modifies salió:

• salió bien = turned out well

Compare:

• bueno / buena = adjective
• bien = adverb

Examples:

• El pan está bueno = The bread is tasty / good
• El pan salió bien = The bread turned out well

So bueno describes a noun, while bien modifies a verb.

Yes, it sounds natural and standard in Spanish from Spain.

Several parts are especially natural:

• salió bien for food turning out well
• tan... que for so... that
• otra rebanada
• para acompañar la sopa

A speaker from Spain would understand and use this kind of sentence without any problem.