The "Wouldn't it be cool if we could do this" principle is a fundamental principle of science fiction.
The point of fiction is that stuff can happen in fiction that couldn't happen in reality - or could, but doesn't. The point of science fiction, specifically, is that stuff can happen which would generally depend on an advancement and application of science and technology beyond its current level. (Hard science fiction can be distinguished from what would presumably be termed "soft" science fiction by stating that hard science fiction depends solely on the judicious re-application of current-day technology. The point where soft science fiction merges into fantasy is a little blurry. Here I think you would have to look at how scientific the underlying principles of your fictional universe are, and decide for yourself.)
But, naturally, good science fiction should also be worth reading. Which means something has to happen in the story which makes you want to read it. And that something should involve the science in some way, otherwise there would be no point in writing science fiction as opposed to just plain fiction.
Something should happen which is cool.
The "method", then, becomes this: dream something up. It doesn't have to be an invention, it can be a concept, an ability, a plot element, an event, an action - anything that takes some kind of scientific idea beyond its usual boundaries. Read stories about new kinds of science and other science fiction stories and generally subject your mind to interesting stimuli until inspiration of this form strikes. Repeat, writing down everything you think of which is interesting until you eventually get something you think you can turn into a good story.
Next, spend as much time and brainpower as you can get hold of turning this new cool idea over, figuring out all its logical consequences and possibilities. Construct the universe into which this idea must fit. Who had the cool idea, and how did they put it into practice? What are the consequences? What are the applications? How can a universe coexist with it, if at all? Who profits? Depending on the length of story you're writing you may or may not have a lot of free space to construct this universe. Depending on how much faith you have in the bare coolness of your idea, you may or may not want to expend lots of text constructing it. I think shorter is better, but that's because I'm always terrified my idea might just fall flat on its face, so I shy away from potentially wasting too much of the audience's time. Going longer, you will probably need more than just one cool idea to keep the reader interested.
Now you should have come up with some interesting consequences of your cool idea. This is absolutely critical. In most cases, it is not enough just to come up with the cool idea. Your story cannot be "Something cool happens!" unless 1) the cool thing happens right at the very end of the story, 2) the cool thing is a surprise, and 3) the cool thing is cooler than the story is long (i.e. the punchline is worth it). You need to do something with the cool idea.
What you do with it is up to you, of course. Beyond this point, standard story-writing advice holds. Grab attention within three sentences. Make dialogue believable in context. Explain exactly as much as you need to...
Now I'd be the first to admit that this isn't the only way to write science fiction. For one thing, this completely disregards the fields of character-driven or emotion-driven science fiction. I generally write shorter stories and getting the reader to have a significant amount of empathy for a character in the story is difficult in a short space of text. Likewise, pulling emotion directly out of science is a very much trickier process than making it cool.
But it's all possible. I'm not telling you how to do it, just one way that I do it, sometimes.