Learn Spanish (Spain): Han anunciado el despido esta mañana. - They announced the firing this morning.

To say "They announced the thing about the firing this morning", you would replace el despido with lo del despido (combining de + el). The structure lo de refers back to a known event or situation without stating it fully, which is perfect if both speakers already know a firing was going to happen.

Why does this sentence use the present perfect (han anunciado) instead of the preterite (anunciaron) for something that already happened?

In Spain, the present perfect is the standard tense to use for completed actions that fall within an unfinished time frame. Because the action happened esta mañana (this morning), it is part of "today", which is still ongoing. In most of Latin America, you would typically hear the preterite (anunciaron) here instead.

Is el despido related to the verb for saying goodbye?

Yes! The verb despedir means "to say goodbye", but in a business context, it specifically means "to fire" or "to dismiss" someone. El despido is the noun form, meaning "the firing" or "the dismissal".

The English translation says "They announced", but there is no ellos. Is this just a dropped subject?

It is likely an impersonal use of the third-person plural. In Spanish, using a verb like han anunciado without a specific subject is a very common way to express that an action was done by an unspecified group or authority, similar to saying "It was announced" or the generic "They announced" in English.