Explaining something as counteractive as quantum mechanics is difficult, and explaining anything to a flat earther is difficult. Combining the two makes for one formidable challenge, so here goes:
The first thing you must realize is that you have to learn to think outside of classical Newtonian mechanics. Your brain is wired to survive on this world we find ourselves in and that does not require understanding the behavior of things as a quantum scale, so expect that this idea will challenge your way of thinking. Everything seems to have a definite position and velocity, behave in a single easy to predict way, and the act of observing something doesn't seem to effect it. When you get down to a quantum scale all these things you think you know are wrong, which seems insane but bear with me and remember that your brain is not wired to understand this.
Quantum objects move in a way that you cannot predict with certainty, you can only say the probability of it being in a certain place at a certain time. Basically the particle "sniffs out" every single potability and then settles on one once it is observed. It should be noted that when I say "observed" I mean that it is interacted with in such a way that it's possible to determine it's position. The different possible locations of the particle form a wave pattern that can interfere with it's self as seen in the double slit experiment, which produces an interference paturn even if the photons, electrons, or any other fundamental particle are fired one at a time. A way of thinking of this that I find helpful is a speck of dirt in a rain drop: at first it's all in one place and then it spreads out like a wave when it hits the ground, the dirt is more likely to be in the crests of the wave because there is more water there.
Another drastic revision that needs to be made to your classical way of thinking is about the certainty in an object's position and velocity. In a nutshell Heisenberg's uncertainty principal states that the more you know about an object's position the less you can know about it's velocity and visa versa. If you wanted to know the position and velocity of, say, an electron you would do that by firing a photon at it and seeing how it's altered. The accuracy in which you can measure the position of the electron with this method is based on the wavelength you fire at it because your accuracy is always within one wavelength of that light. The problem however is that wavelength is inversely proportional to the energy of the photon and the higher the energy is the more it disturbs the velocity of the electron. All other experiments you could possibly preform on the electron are like this, it's not just a technological limitation but a fundamental law of the universe. We know this because shooting light through a slit and making the slit thinner will make the light smear out when it gets thin enough because the small slit means that the low uncertainty in position (because we know that the photon is in that small slit) will make the uncertainty in velocity rise, causing the light to change directions and smear out.
Something that is commonly associated with quantum mechanics is the fact that things can have multiple states and collapses into one when observed, I will explain that in more detail. The double slit experiment is the perfect example of this, because a particle goes through both slits, one slit, the other slit, and none of the slits at the same time as the probability wave passes. The path that the particle finally settles upon is undefined. If you try to measure a particle to see which slit it goes through then that will disturb it enough to collapse the quantum wave form and then the particles behave like particles and hit the screen in a way that you would expect if you put particles through the double slits, but removing the measurement devices causes the interference pattern to return.
Quantum entanglement is another interesting topic. There is a property of all particles called "spin", the particle is not technically rotating but the analogy is still a good one. Every particle has a set spin "speed" that's intrinsic and impossible to change, however the direction of the spin can change. Photons have a spin of one and if you create two photons together in a process you know will produce photons with a combined spin of zero then those photons are now entangled. Being quantum objects their spin is uncertain until measured, and so they are a superposition of both spin up and spin down at the same time. If you measure one of the photons and it's spin up then you immediately know that the other one is spin down without measuring it which means that it's superposition has been collapsed, and if it's measured it will indeed have spin down. Basically one photon collapsed the superposition of another without communicating with it, and this doesn't violate spacial relativity because classical information such as a message cannot be transmitted instantaneously this way. You could argue that the spin of the photons was decided at the moment of their creation, but there are statistical experiments that have disproved this.
Anyway, if you have any questions feel free to post them, although I probably didn't have to tell you that. I tried to be as concise as possible and explain this in simple terms, so this does not represent all that people know about quantum mechanics.