Learn Italian: Fanno installare la luce all'elettricista. - They're having the electrician install the light.

This is the core rule of the fare + infinitive structure! If the second verb has a direct object (here, la luce), the person performing the action must become an indirect object. Because you can't have two direct objects, the electrician gets the preposition a (becoming all'elettricista). Think of it literally as: 'They make install the light to the electrician.'

Why does installare come directly after fanno? Can I say 'Fanno l'elettricista installare la luce'?

No, you can't separate them. In Italian causative sentences, the conjugated form of fare and the infinitive verb (like installare) act as a single unbreakable block. You must keep them together, unlike in English where we put the person in the middle ('have the electrician install').

Wait, who is actually doing the installing here?

The electrician is doing the installing. The subject (an unstated 'they') is just causing the action to happen by hiring or asking the electrician. In this structure, the person introduced by a is the one who physically performs the infinitive verb.

As a quick refresher, how is the word all'elettricista formed?