Inside the Phone equipment room the other end of those two 26 guage wires meet up with a DPDT relay with a coil voltage to match the 2 relays of 12 volts DC, and are wired to the Common switch terminals of the DPDT relay, one wire to each pole.

The two poles of the DPDT relay, have their contacts cross connected
    ( N.O. to N.C. ) so that one polarity is present  at the two common switch contacts with no relay coil voltage, and then the reverse polarity is present when the coil voltage is applied, and the contacts switch on both poles.

The DC power supply is wired to the Normally open contacts, one Normally open contact has the - polarity, the other has the + polarity, this + polarity is also wired to one of the coil wires. A Cathode of a 1N4002 spike absorbing diode also goes here, the Anode of the spike absorbing diode goes to the other coil wire. The other coil wire goes to one of the Business Phone equipment switch contacts, the other related switch contact of the same Normally Open switch goes to the - polarity on the DC power supply.

The other Business Phone equipment switch contact has one connection to the
 ( - polarity)  on the  ( - Normally Open contact ) on the DPDT relay, then remove the cross connection wire  from this Normally Open contact, and connect the other related switch contact to the Normally Closed contact that the cross connect wire was connected to. Now when this Phone system contact is closed, the polarity at the Normally Closed contacts will energize the two wires, and thus energize the coil of one remote relay.

The other Phone system contact will energize the DPDT relay coil and this DPDT relay will transfer its contacts and present the opposite polarity to those same wires and energize the other remote relay.

The 2 wires to the Remote relays have no power until a Business Phone System contact closes.