La dentista me dijo que tenía el labio un poco hinchado, pero que se me pasaría pronto.

Because the speaker is referring to a female dentist. In Spanish, many profession nouns change article according to the person’s gender:

• el dentista = the male dentist
• la dentista = the female dentist

The noun dentista itself usually stays the same; the article shows the gender.

In Spain, dentista is very common in everyday speech. You may also hear odontólogo / odontóloga, but dentista is the more usual everyday word.

Me means to me.

• me dijo = she told me
• dijo = she said / told

Spanish often uses an indirect object pronoun with decir when saying who was told something:

• La dentista me dijo... = The dentist told me...

So me is necessary here because the person receiving the information is part of the meaning.

Tenía is the imperfect of tener. Here it means I had in the sense of my lip was in that condition at that moment.

Why imperfect? Because the sentence is reporting a state or condition, not a completed action:

• me dijo que tenía el labio hinchado = she told me that my lip was swollen

Using tuve would sound wrong here, because tuve is a completed event, not a continuing state.

Why not estaba? You could also hear:

• tenía el labio hinchado
• tenía el labio un poco hinchado
• tenía el labio inflamado

This is a very natural Spanish pattern: tener + body part + adjective.

So tenía el labio hinchado is a normal way to say my lip was swollen.

With body parts, Spanish very often uses the definite article (el, la, los, las) instead of a possessive like mi or tu, especially when the person affected is already shown by a pronoun.

Here, me already tells us whose lip it is:

• La dentista me dijo que tenía el labio hinchado
• literally: The dentist told me that I had the lip swollen
• natural English: The dentist told me that my lip was swollen

This is extremely common in Spanish:

• Me duele la cabeza = My head hurts
• Me lavé las manos = I washed my hands
• Tiene los ojos azules = He/She has blue eyes

So el labio is exactly what you would expect here.

Because hinchado agrees with labio, and labio is a masculine singular noun.

• el labio → hinchado
• la cara → hinchada
• los labios → hinchados

Adjectives in Spanish usually agree in gender and number with the noun they describe.

So:

• el labio hinchado = the swollen lip
• la encía hinchada = the swollen gum

Un poco means a little or a bit.

So:

• hinchado = swollen
• un poco hinchado = a little swollen / a bit swollen

It softens the statement. The dentist is not saying the lip was badly swollen, only slightly.

Other similar expressions are:

• un poco = a little
• algo = somewhat
• bastante = quite / fairly
• muy = very

Because the sentence contains two clauses both depending on me dijo:

• me dijo que tenía el labio un poco hinchado
• pero que se me pasaría pronto

That second que links the second reported idea back to me dijo. In English we often omit that, but in Spanish repeating que is very common and natural.

So the structure is roughly:

• She told me [that X], but [that Y].

Without the second que, the sentence would sound less natural in this kind of careful reported speech.

Here pasarse means to go away, to wear off, or to stop.

So:

• se me pasaría pronto = it would go away soon

In context, it refers to the swollen lip.

This verb is very common for temporary problems:

• Ya se me pasó el dolor de cabeza = My headache has gone away
• Se me pasará = It’ll pass / It’ll go away
• Se te pasará pronto = It’ll go away soon

So this is not the basic meaning of pasar as in to pass by. It is the pronominal use pasarse, which often refers to something temporary ending.

This is a very common Spanish pattern with things that happen to someone, especially physical sensations, problems, or temporary conditions.

In se me pasaría:

• se is part of the verb pasarse
• me shows the person affected: for me / to me

So:

• se me pasa = it goes away for me
• se te pasa = it goes away for you
• se le pasa = it goes away for him/her

You do not normally translate both words separately into English. The whole expression means something like:

• it goes away
• it wears off
• it passes

with the idea that I am the person affected.

Because this is future from a past point of view.

The main verb is in the past:

• La dentista me dijo... = The dentist told me...

So what she said about the future is often shifted into the conditional in Spanish:

• me dijo que se me pasaría pronto
• literally: she told me that it would go away soon

This is like English reported speech:

• direct: Se te pasará pronto = It will go away soon
• reported later: Me dijo que se me pasaría pronto = She told me it would go away soon

So pasaría is very natural here.

Yes. That would also be natural.

• me dijo que se me pasaría pronto
• me dijo que se me iba a pasar pronto

Both mean roughly she told me it would go away soon.

The version with the conditional, pasaría, is a bit more compact and is very common in reported speech.
The version with iba a + infinitive can sound a bit more conversational.

Not exactly.

• pronto here means soon
• rápidamente means quickly

So:

• se me pasaría pronto = it would go away soon
• se me pasaría rápidamente = it would go away quickly

Those are similar, but not identical. Soon focuses on when it will happen; quickly focuses on how fast it happens.

In this sentence, pronto is the most natural choice.

Yes, but told is usually the best translation here because the sentence includes the person who received the information:

• La dentista me dijo que... = The dentist told me that...

If you say said, English often prefers said to me:

• The dentist said to me that...

That is grammatical, but less natural in everyday English than told me.

So the Spanish is literally based on decir, but the best English translation in this sentence is usually told me.

It breaks down like this:

• La dentista = the dentist
• me dijo = told me
• que tenía el labio un poco hinchado = that my lip was a little swollen
• pero que se me pasaría pronto = but that it would go away soon

So the pattern is:

[person] + me dijo + que... + pero que...

This is a very useful structure for reported speech in Spanish. For example:

• El médico me dijo que todo estaba bien, pero que tenía que descansar.
• La profesora me dijo que el examen estaba bien, pero que podía mejorar.

So the sentence is a very good example of how Spanish reports what someone said in the past.