I know there is a difference between 'hard' and 'soft' science fiction. But even then, it can be hard to know what sort of categorization in which to place and treat your story, and it is not really clear how little science 'soft science fiction' is allowed to have.
You only need to present as much information as is necessary for the plot.
Let's use the classic SF technology, Faster Than Light travel. If all it's doing in terms of the plot is moving the characters from A to B, no further elaboration is necessary. On the other hand, if the method is important, you need to get into it more. To use an example I've read recently, Glynn Stewart's Starship Mage series. There FTL is literally magic; a specially trained mage can cast a spell that teleports a ship no more than a light-year in a single jump. Because it's important to the plot, you quickly learn more about it and the limits; how often a mage can make a jump without their head (literally again) exploding; how the ship has to be prepared for the spell to be cast, details about the special place on the ship where the mage operates, and so on.
Because the series features space battles and pursuits and piracy and boarding actions and all the classic space opera fun, all of that is relevant. You need to know a bit about how the (magic) tech works and its limits to justify why, for instance, the main characters can't simply jump somewhere else when the bad guys show up, or why seizing the ship's sanctum means they can't run, and so on.
Now compare that to, say, Star Wars. You need to make calculations to jump to hyperspace, you can only fight in normal space, and in some places you can't go to hyperspace. That's about it. But in terms of the story, that's all you need to know.
You should think about the science in Science Fiction in the same way that you think about everything else in your story:
Is it relevant to the story?
- Yes => Include it
- No => Don't include it
In soft SF, worldbuilding and description are subordinate to storytelling. In hard SF, storytelling is a means to transport ideas or facts.
When Arthur C. Clarke wanted to inspire us to build a space elevator, he included all the necessary technical detail in The Fountains of Paradise. When Ann Leckie wanted to make us think about language and gender, she didn't distract us with details of how AIs control human bodies in Ancillary Justice. Clarke has been criticized for his flat, unemotional characters and praised for his vision; Leckie's novel has received acclaim as a character study, but was criticized for its simple ideas.
Some stories explore the impact of technological discoveries or achievements on society or mankind at large. That could be FTL, a new weapon, a cornucopia, nanotech or simply a space elevator. These stories have tech at their center and need to provide more detail about their characteristics. If we want to explore near-future developments which we consider realistic we must also provide a plausible scientific background and path leading to the technological achievement.
Other stories, by contrast, just use SF as a setting for entertainment or exploration of the human condition; there, technology is just a vehicle for the story and is not at its center in the same way and therefore doesn't need the same level of detail and explanation.
As an example from a different genre consider the role of weapons in a Western. If we are just telling an adventure in a Western setting it's enough to know that you can shoot people with a gun at some distance. If, by contrast, we are telling a story about the impact new weapons like the Winchester rifle or machine gun had on politics, economy and society of the American West in the late 1800s the specific characteristics, means of production, cost etc. of these weapons will be an intrinsic part of the story.
I believe that in any case of Science-Fiction, you 'need to know' more about the science you are working with than you have to tell in the story. This is mostly to make sure that your fictive stuff makes sense in connection to each other; it abides by the same rules, etc.
Put in other words: The better you understand the tech you've made up, the better you can portray it authentically, be it light-speed travel, teleportation, or anything else.
You don't necessarily need to provide your justifications to the reader but you should always have a specific and internally consistent justification for how your setting, and its technology, works. Such a framework will keep your story(s) consistent and control any tendency towards supertech runaway where you just throw more tech at problems instead of working within an established set of rules.
As to how much you do tell the reader, that's ultimately up to you but respect your beta readers when they say we need more information or X doesn't make sense, include the information that is directly relevant to the plot and as much more as you feel is warranted.
To add another twist to the "is it relevant?" answer, consider the tech in any contemporary fiction.
Does it include people driving anywhere? If so, do they describe the internal combustion engine, or even describe the car? If they've got a plane somewhere, how about theory of flight?
Historical fiction, you're more likely to have some description of non-contemporary items. Still, do they describe how to make a candle or a rushlight?
What matters is the purpose it serves. If its behaviour needs to be described for a plot point, then go for it. if there's a car chase scene then the car will always be described, because it matters whether the hero is driving a Lamborghini, or a Ford Focus. But still, what matters is how the car handles, not describing the engine inside it.
I'd recommend providing as little information as possible. It leaves readers to speculate on how things work and provides you with no limits in the future to what things can or can't do. I'll reference Harry Potter. Use of multiple wands by 1 person, where does the energy come from for spells, limits to how powerful a spell can be, why everyone just doesn't use Avada Kedavra to immediately end fights, why elf magic is so powerful, etc. None of these issues are ever addressed in the books, yet people love to speculate and discuss these issues and the books are huge successes. Then take the Eragon series. The author does a great job of explaining where the energy comes from in magic, explains limits to everything, but the writing gets very detailed and limits what the characters can do in magic since it isn't limitless like in HP. The eragon series is successful, but no where near HP.
It took 10 years to write 7 HP books. 9 years to write 4 Eragon books. It's your world you're writing. You set the rules, not the other way around.
Issac Asimov, widely regarded as one of the fathers of the science fiction genre, didn't seem to believe details were any more important than you chose to make them. As Ubik has quoted him here: https://scifi.stackexchange.com/a/51327, he chose the term "positronic brain" for little reason more than it sounded cool and futuristic. This is from the author whose novels shaped our collective expectations of robots, and he essentially said "I care more about how robots would affect the world than about how robots work." I feel that's solid advice for a writer: if it doesn't pertain to the subject matter of the work and it isn't blatantly breaking the laws of physics in ways that ruin immersion, then you don't need to bother with it.
"How important is the science in science fiction?"
It depends upon the audience...If you are writing a hard science fiction story where the protagonist is an engineer who uses her skills to keep her crew mates alive by re-purposing ship board machinery on their crashed ship...You will be explaining or showing why it worked.
However, in most cases do not give in to the temptation to over-explain the science. Most of us know what an internal combustion engine is. However, if you walked around the streets of a typical metropolis and asked passers by to describe the scientific principles, let along the math, that is use in the design and manufacture of that engine, over 95% of respondents could not accurately describe how the engine works. For most drivers, all they need to know is how to operate the thing...not how to design and build it.
Unless the science is critical to the story, most readers do not need to know the "nuts and bolts" of how the science works. Most characters who interact with the technology are users...not maintenance or designers. Show it working and maybe how the character uses it...don't jump out of the story and go into a ten paragraph physics lecture on the scientific principles behind its use.
A way to think about how much detail you will need is to think of driving a car. For most of us, we get inside, fasten the seat belt, insert keys into the ignition, turn it on, press the brake pedal, look around, shift the lever into drive, release the brake pedal, and press the accelerator, turning the wheel to control our directions.
99.9% percent of us, do not think about the interplay between the battery, starting motor, engine, transmission, steering, suspension, tires, and the host of other parts. Unless something is wrong, we are focused on getting where we need to go. Unless something goes wrong, we don't think about all parts and their roles. Likewise, most characters will focus on what's important to them at that moment.
Even mechanical engineers who design cars do not think about the interplay of parts on their drive into work. They are more focused on getting to work. They know about all the science and technology, but that is not what is important at the moment. Focus on what is important and your writing will be better than 95% of the stuff that many authors try to pass off as science fiction.
Just to add to the other (very good) answers, it's worth pointing out that if your story is from the point of view of an inhabitant of that world, they might not know how a lot of things work.
Imagine a contemporary novel, where someone rides an travellator (a horizontal "escalator"), something that might seem magical to someone from the 18th century. Does our protagonist know how a travellator works? Probably not. They could have a guess perhaps. But, either way, the novelist doesn't feel the need to explain how it works.
In addition, if the technology is not well understood by you, the writer (which is inevitable if you're talking about something like wormholes, where nobody really knows how they might work, even their "inventor" Kip Thorne), then the more details you give, the more there is to pick apart and find flaws. If you just say "Sven took the Cannula and was on Tranquility base within minutes" then the reader's imagination has to do more work.
Another reason why you should go as light on the science as you can is to avoid jarring knowledgeable readers. Suppose you need a faster-than-light drive for your story. Cool, I have no problem with that. Now suppose you try to explain it scientifically. I probably know enough physics to realize why what you said wouldn't work, and that will be unnecessary verbiage that will jar me while reading.
Good science fiction is good literature. Therefore, the answer is the same for any literature. Sophistication messes the flow and the flow state of the reader, more importantly. Any sophistication you bring should be worth the high cost just mentioned. One case the answer is obvious: If the sophistication adds to the beauty of the text and the message/meaning of the novel it is definitely worth the cost.