Solar flares represent vast releases of energy in the form of high energy charged particles shot forth from the sun. This radiation will damage most anything, but some things are much more vulnerable to it than others.

Since the particles are charged, they induce electric fields. Anything designed to detect electric fields (such as antennas of any kind) will be interfered with for this reason. The electric fields, if strong enough (there are enough particles, or the particles are energetic enough), then they will induce currents in ordinary wiring, which can fry electronics. An EMP wave (Electro Magnetic Pulse) from a nuclear blast or as used in various science-fiction stories and movies will do the same, for the same reason.

The earth's atmosphere acts as a shield against these particles, as does the earth's magnetic field (it channels them into the poles, where they cause Auroras). Consequently, these pulses of particles are most damaging on things in space, such as global communication satellites. The output of solar flares is usually not enough, however, that astronauts get radiation sickness from the brief exposures that they have in space. These particles from the sun do interact with the atmosphere, however, producing the radioactive Carbon 14. Between ten and twenty percent of the radioactivity that we humans are subject to on a daily basis comes from these radioactive isotopes in the atmosphere made by the sun (most of the rest is radon released from radioactivity in the earth, and atmospheric effects from cosmic rays from outer space. Nuclear power/weapons is actually less than a percent).

Solar flares are storms of magnetic energy. Here on Earth, they mostly affect us by interfering with cell phones, causing static on the radio, and causing blackouts. Satellites, however, can experience a lot more damage because of solar flares. Solar flares send electrical currents through the satellites that damage sensitive electronic equipment inside.