If you’re going to write in the science fiction or fantasy genres, sooner or later you’re going to end up handwaving an explanation. Other genres sometimes do it too, but other genres don’t regularly work with props and plot elements that don’t yet and may never exist.
Some handwaving is easy, because the genre as a whole expects it. Take faster-than-light space travel, for example — sf writers have been handwaving that one for so long that all they need to say is something like “hyperspace” or “wormhole jump” and the reader is there for the ride. Readers aren’t dumb. They know perfectly well that if the author of the story actually had the plans for a working faster-than-light drive, he or she wouldn’t be writing adventure stories for a living. Too much explanation, in this case, would bring on a case of Handwaving Fail — all the audience wants to know is that the author is aware of the problems with faster-than-light travel, and that for the purposes of the story, those problems have been solved. They don’t really want all the equations plus a diagram.
Sometimes the handwaving has to be subtler. If you’re introducing a bit of tech that you’ve postulated just for the occasion, don’t draw attention to its extraordinary or purpose-built nature. If you talk about it and around it as though it’s been hanging around the laboratory or the workbench or whatever since well before the current problems started, the reader will think of it as just another piece of the furniture, and with that you can slip it into its place in the story without occasioning comment.
If you’re looking at the need for a really large job of handwaving, stronger measures may be required. My husband and I once co-wrote a YA thriller for a packaged series, and found ourselves working with a plot that would have ground to a shuddering halt if the main character had ever actually sat down and talked to the police about what was going on. We got past that difficulty — and did it without making our main character an idiot — by compressing the timeline of the novel into two or three days and keeping our protagonist on the move and short on sleep the whole time.
So. Three general tips:
Use built-in handwaves where the genre allows for them.
Don’t point at what you’re handwaving.
And when in doubt, keep things moving fast enough that nobody has time to stop and think.