Maybe it will help to try and extrapolate from something intuitive.
When cresting a hill on a rollercoaster, you will often feel "Zero-G", where you feel no force from anything touching you (i.e. you lift off the seat slightly, but aren't being pulled down by the restraints either). Obviously the roller coaster isn't somehow turning off gravity for you alone, so what's happening to make you feel weightless?
What we often refer to as "Zero-G" actually has another name: free-fall - the difference is that when you jump out of a plane, you know you're falling. When you're in a roller coaster your perception will often be a little bit different; because the vehicle is moving in relation to the Earth (i.e. falling), but you're not moving in relation to the vehicle, it may feel a bit like when you're floating in water - you're not moving, but nothing is actively keeping you in place.
This effect is achieved through careful engineering, such that the parabolic arc of the track follows the parabolic arc of gravity. Notably, this is the exact same arc that a canon ball would follow, if you launched it at the same speed as the roller coaster.
Now, instead of thinking on rollercoaster speeds, let's go a little faster - if you imagine the parabolic arc of your roller coaster hill, and add more horizontal speed (but kept the vertical speed the same), the parabola would stretch out wider and wider. Eventually, it would reach the point that the two ends (the points where the parabola intersects the Earth) would actually be behind the horizon from each other - and at that point, something interesting happens: you're going so fast horizontally that the amount of time it takes you to reach the ground increases, because the ground is actually pulling away from you a little bit, due to the curvature of the Earth (being a sphere).
Now add even more speed! The faster you go, the further apart the ends of your parabola are going to get - at least until your parabola covers the Earth from end to end, allowing you to start at the South pole and land at the North pole. But what happens if we go even faster? The ends are going to continue to get farther away from the highest point of your arc (which could now reasonably be called an "apoapsis"), but will start getting closer to each other, towards the other side of the world.
Now you keep adding more and more speed, and pushing the ends of your parabola closer together until they touch... And you're in orbit. Any speed you "lose" on your ascent will be regained after passing your apoapsis, and you'll end up at the bottom going the same speed you started at, meaning you'll keep going around and around, never touching the ground.
If you add just a little bit more speed at this point, you can raise the bottom (now called "periapsis") to be equal to your apoapsis, and you'll have a circular orbit: you never get closer or further away from the Earth, and you'll just continue orbiting forever.
And all it takes is to fall sideways really fast.