Learn Spanish (Latin America): Curiosamente, el jefe lo despide y él no se despide de nadie. - Interestingly, the boss fires him and he does not say goodbye to anyone.

This is a classic example of a verb changing its meaning when used pronominally (with a reflexive pronoun). As a standard transitive verb, despedir means 'to fire' or 'to dismiss' from a job (as in el jefe lo despide). But when you add the reflexive pronoun, despedirse shifts to mean 'to say goodbye' (as in se despide).

Why do we need the preposition de after se despide?

The pronominal verb despedirse specifically requires the preposition de to indicate who you are saying goodbye to. It translates roughly as 'to take one's leave from someone', which is why se despide de nadie means 'he does not say goodbye to anyone'.

In the phrase lo despide, why do we use lo instead of le?

The verb despedir (to fire) takes a direct object, because the person being fired directly receives the action. In Latin American Spanish, the direct object pronoun for 'him' is lo. Using le here would be an indirect object pronoun; while common in Spain (a regional feature called leísmo), it is generally not used this way in Latin America.

Why does the second half of the sentence use both no and nadie?