Learn Spanish (Latin America): Lo importante es delegar el entretenimiento a los vecinos. - The important thing is to delegate the entertainment to the neighbors.

The structure lo + [adjective] is a shortcut to say 'the [adjective] thing'. You only need lo que (the thing that / what) when you are introducing a verb phrase. You could rewrite this sentence as a full pseudo-cleft using lo que es importante (what is important), but lo importante is a much more natural and common way to express this idea.

Why do we use the infinitive delegar right after es?

When you define an abstract concept like lo importante with an action, Spanish uses the infinitive to act as a noun phrase. It translates directly to 'The important thing is to delegate'. You would only need the subjunctive here if you introduced a new subject with que, for example: Lo importante es que tú delegues (The important thing is that you delegate).

Why do we use the preposition a in delegar ... a los vecinos?

The phrase a los vecinos acts as the indirect object of the sentence, as they are the ones receiving the task. Note that you will also frequently see delegar en alguien, which emphasizes placing your trust 'in' that person to handle the responsibility.

Could I use diversión instead of entretenimiento here?