In several press conferences, employees of NASA or private space firms have been asked if they played KSP, and some answered with "Yes".
NASA used patched conics to find candidate orbits for Apollo back in the days.
With that being said, KSP strikes the balance between accuracy and simplicity. Patched conics give a good idea how space works, without being so overly complicated that it's no longer fun to deal with. And that means it makes people learn stuff they otherwise wouldn't. The mantra "Space isn't high up, it's about going sideways fast" can be repeated thousands of times, and people won't get it. But when they actually try it out and see for themselves why it is how it is, they suddenly understand. When they fall back onto the planet after launching straight up, and then finally learn to pitch over and do a gravity turn to achieve orbit.
The fact that orbits are always influenced at the opposite side of the burn. That deceleration (retrograde burns) lowers the orbit and so forth. What radial burns do, what normal burns do, what inclination is and why it is important, how ascending and descending nodes work and why they are important, why rendezvous takes so long and what you need to do to actually make it. All those things are taught by KSP and make spaceflight accessible to people that otherwise wouldn't "get" it.
The Mun (the equivalent to the Moon in KSP) is in an equatorial orbit to make it easier for new players to get there (no launch window, no inclination change), but those things can be used to get to the second, much smaller moon, Minmus.
The concept of Hohmann transfers, bi-elliptical transfers and other transfers is explained. The difference between fuel-efficiency and time efficiency can be learned by KSP. Ejection angles and launch windows for interplanetary missions can be learned with KSP. Gravity assists work, and can be very useful. That's a concept a lot of people struggle with, but in KSP you can try it out and see how that works.
Re-entry heating exists and matters somewhat, depending on the settings. You learn that you should neither plow too fast too deep into the atmosphere neither too low so that you skip it. The concept of re-entry is hard to grasp for some people, but with KSP, you can easily see why it works how it works and what's important about it.
The rocket equation, and especially the tyranny of the rocket equation, is important in KSP. You learn good engineering techniques. You learn that you can't bring what you want, that rockets grow exponentially. The concept of staging and why it is important comes naturally with KSP. Some basic aerodynamics — how CoM and CoL work, how drag affects the path of your rocket — are explained.
What Isp is and does is explained. That Isp at sea level and Isp in vacuum are different things and are important is explained. That you can steer a rocket with gimbaling engines, reaction wheels or RCS is explained, and that some of those are better in some situations then others (RCS for ascent? Not the best idea!).
In short: KSP teaches a lot, while using physics and orbital mechanics that are not so overly complicated that it gets too frustrating. It stays in the area that most people can still grasp.
And for those who want the extra challenge, the game can be modded. N-Body physics are provided by a mod called "Principia", and the star system can be transformed to the Real Solar System by the mod of the same name (RSS). Historic engines and other parts are provided by the mod called "Realism Overhaul" (RO). Signal delay is provided by RemoteTech, life support by one of the various LS mods (Kerbalism, USI-LS etc).
So apart from the base experience that already teaches lots of the concepts of spaceflight (I forgot about heating and radiators in my list above) while maintaining accessible and approachable (if you don't want to learn too much, trial and error will get you somewhat far), the modding community has created a lot of mods that bring even more important concepts to the player (FAR has a more realistic aerodynamic simulation, for example).
And the game lets you deal with it how you want. Don't want to learn about launch azimuth? Fine, don't. The parts have enough delta-v so that you can be sloppy sometimes. Want to learn? Good! You can learn about it, and then apply what you have learned and see it actually work. See your orbit end up exactly what you wanted (or not). The community has created lots of maths tools and helpers for that as well.
Finally, Squad (the developer) already has/had a deal with NASA. NASA helped them make some parts, the so called "NASA parts" that are still in the game today (I believe they were originally for the "Asteroid day" official mod, but are now in the base game).
So yeah, it's not a 100% accurate simulator. But its close enough to teach the concepts, and simple enough so that its accessible for a very wide audience. In that regard, I think it offers a great example for how educational games should work in the future. Because it's actually fun to play and makes stuff easy to understand.
Some random things I haven't mentioned: Aerodynamics (Rudder, Elevons, Trim, placement of wings, CoM, CoL and concept of lift for planes), how an FDAI (called Navball in KSP) works, reference frames (the FDAI has "surface" mode [rotating reference frame of the body] and "orbital" reference frame [non-rotating frame of the body], as well as "target" frame [frame centered around the own vessel]), connectivity of antennas and signal strength (inverse square law), solar power output (again, inverse square law, you learn why solar panels don't work well for outer planets), you learn about the pendulum rocket fallacy and a lot more things.
On accuracy and computing power:
Being a game, the simulation has a hard realtime requirement. Patched conics are not only far easier to understand for a wide audience, but they are also much faster. Patched conics have an analytical solution. That means it's cheap to calculate positions. Furthermore, finding closest approach / intercepts is cheap, which is needed for maneuver planning. The added benefit is that it makes finding a rendezvous something accessible for a wide audience. It's not fun to try to find a close approach for two vessels when you need a simulator to make that decision. The base concepts apply whether you use patched conics or not. A more realistic model does not add much educational value, but a lot of frustrations for players — who mostly have no background whatsoever in orbital mechanics. Furthermore, the game needs to be able to simulate literally hundreds of vessels at the same time — something that is only possible with patched conics in real time. The aforementioned mod for n-body physics quickly breaks down after collisions of vessels that generate lots of debris.
Furthermore, it doesn't matter whether a vessel is in atmosphere or not from a physics point of view. The physics engine still needs to simulate all parts and all joints between all parts to figure out how the vessel behaves, especially under thrust.
Those hard realtime requirements put a hard cap on what you can realistically achieve on the average home computer. Again, it's a game, it needs quick results, it's not an academic simulation. It's close enough to teach the concepts. It doesn't claim to have 100% scientific accuracy.
To the point raised in other comments, that it uses "17th century maths". Sure it does. But so do a lot of people. You don't start teaching people physics by jumping into quantum mechanics. You don't even start with GR/SR. You start with classical, Newtonian physics. Those are still around, and still valid as ever, and still have their applications. Just because we nowadays know that they aren't accurate on certain scales doesn't mean those physics are useless.
Again, KSP has to strike a balance between accuracy and simplicity. Be accurate enough to actually use the real concepts of spaceflight, while being simple enough to appeal to a wide audience.
Have you ever been at a planetarium? When they show the orbits of planets in the solar system, they use patched conics. Because that's enough to give people an idea how that stuff works, without being too over-the-top. Have you ever looked at orbits plotted in a Jacobian frame? Those look extremely confusing, while arguably being more "accurate".