What makes it possible for the planets to continue to orbit the sun without falling off their orbits?
They are traveling at a speed which basically balances them between their speed and the combination of their own gravity and that of the suns which wants to pull them together. They are falling towards the sun but because of their forward speed they keep missing the horizon of the sun by their orbital distance from the sun. This results in a roughly circular or elliptical orbit. There is nothing there to slow them down and hence they just keep going around and around.
There's no easy answer, except that, astronomers know from use of a wonderful process called science, in no uncertain terms, that for thousands of millions of years large objects with great mass, such as the planets, actually can and do "fall" continuously around objects of considerably greater mass and gravity, such as the nearby star we call the Sun. The motions are stable, somewhat flattened circular orbits known as ellipses. Without the need to go into the physics, the mathematical details, just note this is something confirmed everywhere we observe. We can see it not only everywhere we look in the Solar System but as far away as the stars and their exoplanets, and even among clusters of galaxies of stars. This is precisely why Moons move in orbits around planets.
Gravity and momentum. Momentum keeps an object moving in a particular direction at a particular speed unless an external force acts on it. Gravity is an external force which, if applied roughly perpendicular to the object's momentum will result in the object's transcribed being curved. If the gravity and momentum are evenly matched, then the trajectory will be curved into a circle or, if they're ever-so-slightly unmatched, an ellipse.
The lack of anything stopping them from continuing to orbit the Sun.
The answer is in the question. Any object in orbit is technically in "free fall". What this means is all the planets in the Solar system, plus evey other object are falling towards the Sun. What makes all of these objects "orbit" however is that they are moving fast enough to keep "missing" the Sun, thus they keep falling in a constant elipse around the Sun instead of just falling into it. And because there is no drag in space, there is nothing to slow any of them down, thus the orbit sustains itself.
I have seen many answers that say the reason for this is momentum but it's wrong, it's angular momentum.
In our universe anything that spins does not want to stop spinning due to angular momentum, but we notice that things do stop spinning in earth, why is that?
Thats because friction stops the object from spinning, but in space there's nothing to do so, so things continue to stay in orbit in space.