El esguince puede curarse con reposo.

Because el here is the article meaning the, not the pronoun he.

• el = the
• él = he

At the start of the sentence it is capitalized as El, but it still has no accent because it is the article.

It can do either, depending on context.

• In a specific context, el esguince can mean the sprain.
• In a general or medical statement, Spanish often uses the singular with the definite article to talk about something in general.

So this sentence can naturally mean something like:

• The sprain can heal with rest if a particular sprain is being discussed.
• A sprain can heal with rest if it is a general statement.

This is very common in Spanish and does not always match English word-for-word.

Yes. Esguince is a masculine noun, so it takes:

• el esguince
• un esguince

There is no special rule here that makes it obvious from the ending, so it is best to learn the noun together with its article: el esguince.

Puede is the third-person singular present form of poder (to be able to / can).

The subject is el esguince, which is singular, so the verb must also be singular:

• el esguince puede... = the sprain can...

If the subject were plural, it would be:

• Los esguinces pueden... = Sprains can...

Because after poder, Spanish normally uses an infinitive.

Pattern:

• poder + infinitive

So:

• puede curarse
• literally: can heal / can be healed

This is the same pattern as:

• puede comer = can eat
• puede salir = can go out
• puede mejorar = can improve

Here, curarse is a pronominal form of curar. In this kind of sentence, it often means something like:

• to heal
• to get better
• to be healed

It does not necessarily mean a literal reflexive idea like to cure itself in a strict English sense.

With illnesses and injuries, Spanish often uses curarse where English may prefer heal rather than cure. So puede curarse is a natural way to say that the injury can resolve or heal.

Yes. Both are correct.

With a conjugated verb plus an infinitive, object/reflexive pronouns can often go in two places:

• puede curarse
• se puede curar

Both mean the same thing here.

This is a very common Spanish pattern:

• voy a ducharme / me voy a duchar
• puede curarse / se puede curar

Sometimes one version may sound slightly more natural depending on style or rhythm, but grammatically both are fine.

Because curar and curarse are not used in exactly the same way.

• curar algo = to cure / heal something
• curarse = to heal / recover / get better

Compare:

• El médico cura la herida. = The doctor heals the wound.
• La herida se cura. = The wound heals.

In your sentence, the focus is on the sprain recovering, not on a doctor actively curing it, so curarse is the natural choice.

Con here means with or by means of. It introduces the method or condition under which healing happens.

So:

• con reposo = with rest / by resting

Reposo is a common medical word for rest, especially in advice from doctors or health contexts.

English speakers often know descanso first, but there is a difference in tone:

• descanso = rest, a break, time off
• reposo = rest in a more medical or physical-recovery sense

So con reposo sounds especially appropriate here.

Because reposo is being used in a general, non-specific sense.

• con reposo = with rest
• not with the rest

Spanish often leaves out the article with abstract or uncountable nouns when speaking generally, especially after prepositions.

If you wanted to make it more specific, you could add one:

• con el reposo adecuado = with the proper rest
• con un reposo prolongado = with a prolonged period of rest

But in the original sentence, con reposo is the most natural general expression.

In this sentence, English would often prefer to heal.

So natural translations include:

• A sprain can heal with rest.
• A sprain can get better with rest.
• A sprain can be healed with rest.

All are possible, but for an injury like a sprain, heal often sounds most natural in English.

Spanish uses curarse quite broadly, so the exact best English wording depends on context rather than on a strict one-word match.