The key component in that circuit is the special Dano switch, it won't work with standard 3-way toggle.

The standard import or Switchcraft style 3-way toggle switches we normally use are On-Both-On, which we wire as Pickup 1-Both-Pickup 2. This works because the in the middle position the switch links to both contacts, in the up or down position it breaks one of the conacts.

The Dano switch is an On-None-On type, where in the middle position it makes no contacts and then the up and down positions it makes a contact.

So, in the above circuit, in the middle position the switch does nothing. The bridge pup runs through the bridge volume which links to the neck negative at the neck volume then the neck pup runs through the neck volume and to output. This links the pickups in series, instead of the more common parallel layout. In the front postion the switch links the ground to the neck volume's lug which grounds out the the bridge pup. In the rear position the switch links the neck volume's neck negative lug with its output lug, thus bypassing it.