Basically, magic and electricity are forces similar enough to influence each other. Strong magic tends to correlate with a strong electromagnetic field. Since each human body produces a certain amount of bioenergy, there's a field that can be detected by science. If your magical aura happens to be strong, that tends to boost the bioenergetic field as well. While a typical human field isn't strong enough to mess with technology, a boosted one often is. The more advanced the electronics, the more vulnerable they are to outside interference.
This effect can be dampened in a variety of ways, but as far as I know, there isn't a way to make it stop completely. Some things that help:
[EDIT 11/18/10: As the conversation continues, I'm coming across new ideas and things I knew but forgot to include, so I'm expanding the list below.]
* Magical energy can cause technology to do things that are bizarre, inexplicable, and considered to be impossible. It is particularly prone to draining or overcharging batteries, altering or crashing computer programs or whole computers, causing clocks to gain or lose time and otherwise change settings, and distorting readings from sensor equipment. Another strong clue is if the exact same action leads to different results. If the tech support or other experts are utterly baffled and insist that what's happening can't be happening or makes no sense by any ordinary logic or system parameters, then magic is a probable cause. Always troubleshoot for conventional hardware or software problems first! Those tend to be more common, and are often more straightforward to fix.
* Plastic insulates against electromagnetism AND magic. So do some other synthetics. (It does not insulate against synthetic energy or technomancy, but that's not relevant for this topic.) Use plastic or other synthetics to separate incompatible forces.
* Silk insulates against magic, but will carry static electricity. It's useful as clothing -- a silk outfit will keep much of a person's magical energy relatively contained.
* Skin contact strengthens magic AND transmits electricity. As much as feasible, avoid touching vulnerable electronic equipment. (Usually it is the technology that suffers, but not always: a lightbulb blowing out in someone's hand is a safety hazard.) Some things that are fine six inches away will misbehave or die horribly if touched directly. Gloves or even a sleeve over the hand also help.
* When the ambient electromagnetic energy is high, the magical interference will get worse. That means when a weather front goes over, or there's a thunderstorm, etc. the chance of magic zapping technology is high. If static electricity sparks are discharging visibly/audibly, that's a warning sign.
* Magical energy builds up over time on location. The more time a magical person spends in a given place, the stronger the magical charge in that area will become -- and far more if it's a place where magic is actually performed. This makes technological malfunctions more likely in that area.
* Objects attune to their owners. Even a fancy bit of electronics such as a computer can, over time, build up something of a resistance to magical interference from its owner. This works better for some people and gadgets than others, but usually enough to be helpful. Objects owned by other persons, or used very rarely, tend not to have this benefit.
* Names and words are powerful. Naming a device, such as a car or a computer, may imbue it with enough identity to resist outside influences; this can help preserve its performance. Tip courtesy of jenett.
* Stone magic is worth trying as a means of absorbing energy, shielding equipment, etc. fabricdragon recommends tourmaline and prehnite. I recommend bismuth in its rainbow crystal form, which is especially good for technomancy.
* Some types of energy work, and especially technomancy, can create enough of an energy barrier to shield sensitive equipment from magical interference, in whole or in part. This usually takes some trial-and-error to make it work, but for a magical person who needs to use electronics regularly, it's worth the attempt.
* Some people have enough control over their personal energy to damp it down and/or pull it in closer to their body. This can reduce the frequency and severity of malfunctions. It is a good exercise anyhow. Other times, someone else may be able to arrange personal shielding that will help.
* A technopath may be able to make a given piece of technology resistant to magical interference. There are various ways of doing this, such as shielding it directly, adding a bit of the owner's magical energy to say "ignore this," or creating a technomagical artifact (usually out of synthetic material).
* For someone with partial or greater compatibility between their personal energy and technology, treating a device as a magical tool may help preserve its performance. This works for some people but not others, so approach with caution. Tip courtesy of jenett.
* Strong emotion (positive or negative, but especially negative) tends to influence magic. It also involves a lot of electrochemical brain activity. This raises the tendency toward technological interference. Try to avoid working with vulnerable equipment when emotionally charged, or at least take a moment to calm down first. Point courtesy of jolantru.
* Attempting to work magic on or through electronic equipment is a bad idea; working magic in the vicinity of electronic equipment is not really a good idea either; unless one is a technopath skilled at interfacing these different paradigms. Some people find that they don't have problems with this, which is nifty for them -- but in general, it tends to boost the rate of malfunction. Unless that rate goes really high, as it does for someone with a very strong field, it isn't likely to be identified as magically related but it is still a small extra annoyance. A statistical blip doesn't have to be huge to be valid, merely has to rise above the few percent usually reserved for margin-of-error.
* Combining multiple solutions may work when a single one is insufficient. Try mixing and matching different things.
When people experience technical difficulties that are not readily explained by science, but match the effects of magical interference, I usually recommend considering aspects of both science and magic in seeking to solve the problem. That means, try to figure out what is happening as best you can, list some possible solutions, test each solution and note whether or not it works. Keep notes if necessary. It may take a few rounds, and bystanders from both disciplines may rant at you for mixing the two, but usually something can be found that will reduce the malfunctions to a level that is merely annoying.