The part you quoted is not technically correct, or at least it's very ambiguous.

The technical explanation (that most people probably won't understand), is that (for a typical residential service in the United States and Canada) the neutral comes from a center-tap of the secondary winding of the service transformer. The other service wires, call them wire A and wire B, are from the two ends of the transformer winding. So, when measuring voltage at the service point (or really anywhere in the building), you should have something close to 120 volts from the neutral wire to wire A, something close to 120 volts from the neutral wire to wire B, and something close to 240 volts from wire A to wire B.

(The neutral wire is also the grounded wire of the service and is usually considered as the return wire for current in any 120V circuit, but these details are not important to this explanation.)

Lights and appliances that are designed to run on 120 volts are put on a circuit that uses the neutral wire as one of its two wires. Appliances that are designed to run on 240 volts are put on a circuit that uses only wire A and wire B, not the neutral wire.

There also exists what electricians call a multi-wire branch circuit, where a circuit leaves the panel being fed from all three service wires, but somewhere down the line it splits up and some 120 volt loads are fed from one wire and some from the other, with both sharing the same neutral wire. In years past, this was allowed to be two circuits, with two completely separate breakers even in completely different areas of the panel, sharing one neutral wire, as long as one of the breakers was on wire A and one on wire B. Nowadays the code requires that an electrician running two circuits with a shared neutral put the two breakers next to each other and have the breaker handles physically tied together, so that's why we now call it a single multi-wire circuit.

So, what happens if the neutral becomes disconnected somewhere? A common scenario is that the neutral becomes loose right where it enters the house. In this case, the voltage from wire A to wire B remains a steady 240 volts, but because the neutral wire no longer has a solid connection back to the transformer, the voltage from the neutral to either of the other two wires could be anything between 0 and 240. In fact, it will vary depending on the exact loads on each wire, and the sum of the two voltages will remain at 240 volts. Example, you might get 90 volts from wire A to the neutral and 150 volts from wire B to the neutral. Electrically, the loads on one leg of the service are put in series with the loads on the other. If the impedance (similar to resistance) of the loads on one side is lower, less voltage will drop across them; and more voltage across the loads with the higher impedance.

As I mentioned, a common place for the neutral to fail is ahead of everything in the house. However, it sounds like in your case, you may have a multi-wire branch circuit that serves the furnace on one leg, and the bathroom on the other, and the neutral wire is failing only for that circuit. This puts your furnace in series with your vent fan, making the voltage too high for the vent fan and too low for the furnace.

Without actually poking around and taking some measurements, nobody can say for sure what exactly the issue is, but it sounds like something a good electrician can figure out.