Electronic devices, particularly devices that transmit or receive radio signals (televisions, radios, etc.)depend on information brought to them in the form of electromagnetic radiation, which normally means light(light of a much lower frequency than you can see), but it can also include various other particles that have electric charge.

Solar flares produce a great number of these particles (electric charges) and spew them at the Earth, causing random signals that the receiver has to try to sort out, the actual message from the garbage that the sun is putting out. It's exactly like trying to listen to somebody when a siren is going off next to your ear- you can't tell the useful information (the other person's speech) from the noise (the siren). Solar flares do exactly the same thing to anything that detects radio light, like cell phones.

Solar Flares indicate the activity of the solar wind, which is a fast streaming of charged particles from the outer atmosphere of the sun. The solar wind is typically bent around each of the planet's magnetic fields(in fact the boundary is the effective boundary of the magnetic field). So high degrees of activity (dense winds) affect the depth of the field and generate substantial radio frequency noise at the earth, as well as bright aurora at the poles. This interference is what effects cellphones.

A secondary effect in long-distance communication occurs when the ionosphere bears the brunt of the excess solar wind-- the ionosphere reflects low to short wave frequencies and these are strongly affected by solar storms.