Learn Spanish (Spain): Ella sueña con vivir en el extranjero.

This is exactly what makes soñar a prepositional verb! To say dream of or dream about, Spanish always pairs soñar with the preposition con. You cannot translate the English of literally here; it is always soñar con.

The English says living, so why does the Spanish use the infinitive vivir instead of viviendo?

In Spanish, whenever a verb comes directly after a preposition (like con, de, or por), it must always be in its infinitive form. While English uses the -ing form after prepositions, Spanish strictly requires the infinitive.

How does the phrase en el extranjero translate to abroad?

The word extranjero can be an adjective meaning foreign, but here it acts as a noun. En el extranjero literally translates to in the foreign territory or in the foreign country, which is the standard Spanish phrase for abroad.

Do I have to include the article el in en el extranjero?

Yes, you need the article here. Because extranjero is functioning as a noun that refers to a general concept or place, Spanish grammar requires the definite article el. Saying en extranjero would sound incorrect.