One of the relative miracles of modern times is the reliability of the phone system. The power goes out fairly often for most people. Sometimes it is only out for a second, but other times it can be out for minutes, hours or even days. Your telephone, on the other hand, is always working (as long as you pay the bill). Why is that?
The article How Telephones Work talks about the simplicity of phones and the telephone network. The article shows you how, with just a 9-volt battery and a resistor, you can create your own intercom system using two normal telephones. A phone will work as long as it is getting between 6 and 12 volts at about 30 milliamps. In other words, it takes very little power to operate a telephone.
Between your house and the phone company's office there is a dedicated pair of copper wires for your phone. Those wires are almost always buried, so ice storms and hurricanes will not cut them. The phone company supplies the power that your phone needs using your dedicated copper pair.
So even if the power goes out in your house, the phone still gets the power it needs through the phone line. And at the phone company office there is an extensive battery system, as well as a backup generator, to supply power during a power failure. If the power goes out, the batteries and generators keep the office fully powered. Therefore, all of the phones connected to the office are fully powered as well.