Specifically, solar flares induce currents in long untwisted pair cables. If the cables have high impedance loads on the ends, high voltages will result, frying equipment.
The telegraph was an example of a high impedance long wire.
In modern equipment, long length high impedance lines are rare. Telephones are one, at around 900 ohms, although nearly all cables are twisted so barely affected. Power lines have an impedance below 1 ohm. Most Comms goes via fiber. Cable TV networks have a grounded shield so unaffected. Small household devices are small, and therefore don't have long lines to get big currents.
I'm sure a solar flare would break some stuff, but the vast majority of devices are more resistant than people expect.
The most vulnerable thing in today's world is probably actually disused phone lines, which are currently disconnected from anything else, so infinite DC impedance, yet very long, and could cause sparks and fires since they typically have paper insulation which is only good for 1000v or so.