The trick is a very basic physical intuition error - one that one should not be blamed for, because it took the likes of Galileo and Newton to realize what was wrong with it, and as an idea it held as orthodoxy for ages and across the world prior to the innovations of their genius. It's much like the Flat Earth - there's actually no blame in entertaining the idea, it only gets silly today when you insist on refusing to capitulate to the evidence and reasoning behind why it cannot be so under the guise of finding the belief in some giant "conspiracy" more appealing on an emotional level. And the physical intuition error here is the idea that in order for something to be moving and stay moving, it needs to have a "power source" - something continually inputting energy into it, all the time, to keep it going, and when that power source runs out of energy, the thing will stop moving.
This, however, is not how our universe actually works to the best of our knowledge, though there's no reason a universe couldn't work this way. Instead, motion, in fact, while it can be and is associated with an energy, called the kinetic energy, but this energy, once imparted to an object, will stay with it until something comes along to take it away, meaning that an object left to its own devices and set into motion, will always keep moving, as far as we can tell.
The reason why it was thought for a long time that motion requires a continuous input of energy is from naive observations of motion in the environment on the surface of the Earth. Here, if you throw a rock, say, it eventually stops. It hits the ground and skids to a stop. You can push a wheelbarrow, get it going, then let go, and it soon stops, if not stopping immediately. Indeed, you can't very easily find something on Earth that won't just move and keep moving of its own accord. But today, we understand this to be the result, not of motion being inherently unsustainable, but rather because there are in fact agencies in the environment which are acting to rob energy from the moving object, like air drag and frictional forces (both the rock and barrow are subject to these). Thanks to these, we have to continually resupply energy to the moving object with some sort of engine to maintain the motion, hence why your car is always burning fuel as you are driving.
In lieu of such forces, though, these items would continue to move, and we can demonstrate this by considering identical objects given identical initial impetuses in contact with differing kinds of surfaces. When we do that, we find that the time it takes each to stop depends on the type of surface, strongly suggesting the stoppage is actually due to the interaction with the surface, and not due to some "inherent" unsustainability of motion.
Of course, the ultimate demonstrative proof of this has only been done very recently with the space program: when we send a craft to, say, Mars, the rockets need only burn to lift it from the Earth and then set it on its way. Once that is done, there is no further rocket burning required (and very thankfully for us, otherwise it'd be FAR harder to do space travel than it is now: think about how much fuel a rocket like that burns in just a few seconds, now remember it takes about 20 MEGAseconds to travel to Mars from Earth, and imagine if you had to burn fuel at that rate for all that time, literally millions of times longer! You'd need a rocket with a fuel tank a million times bigger.).
So, finally, we get to Bohr's model. Bohr's model imagines the electrons as free particles in the vacuum of space ("inner space", not outer space, in a way) following Newton's laws of motion, and not subject to any dissipative forces like with pucks and wheelbarrows grinding on hard terrestrial surfaces. Thus, their motion requires no ongoing input of energy to maintain itself.
Add: I see you also inquire about some other features of Bohr's model, like what it means to "have an energy level". What this refers to is the quantization condition. The reason for this requires us to once more think about what I just said regarding dissipative forces. You see, the model that Bohr's was based on, the so-called "Rutherford model", imagined the atom as essentially identical to a miniature solar system, with the electrons being like "planets" and the nucleus serving as the "Sun", and the gravitational force has been replaced by the electric force as the appropriate central force confining things to orbits. However, the theory of electromagnetism, as finalized by James Clerk Maxwell, does predict the existence of a kind of dissipative force in this situation: namely, the emission of electromagnetic radiation, like a radio transmitter. The atom should "broadcast" away all its energy until there's none left, the electrons' orbits decaying like a satellite experiencing atmospheric drag, and finally crashing and burning in the "nuclear Sun" at the center.
Yet clearly, we don't have that, we have stable, interactive and lively atoms, not shriveled, neutral and inert, worthless nuggets of electron-dapped nucleum! Based on this, and some other experimental data, what Bohr suggested was that the orbits must be constrained somehow: and the way he did this was to suggest that the electrons are only "allowed", for some reason that he did not know, certain levels of orbital mechanical energy (not quite, but to keep things simple), and these levels are isolated, fixed numbers, not a continuum of possible values. Moreover, there was an absolute minimum, nonzero orbital energy for each electron, which would ensure that the electrons would never collapse to the bottom. Note that this also requires a violation of EM theory because effectively you have a contradiction in that even with this constraint, the electron still must be orbiting, so what is preventing it from radiating? Hence it's not a particularly satisfactory model, but this is how it works.
The phenomenon of discrete energy levels is, however, an actual empirical fact, which comes from spectroscopy - that's why Bohr used it, so given the above, it must have a different explanation, and this is what led to the development of the fully modern theory of quantum mechanics. In this model, considerably more radical changes to physical theory are required - in effect, we introduce a parameter $\hbar$, which acts as a sort of (in a very rough sense) cosmic "resolution limit" - think about a computer which stores numbers with finite precision, but done more eloquently and subtly - that limits the amount of information allocated to describe the parameters of particles such as their positions, velocities, rotational speeds, and otherwise, and done in such a way that the permitted energy values end up also as being discrete, yet from a powerful, general set of principles that can be used to accurately describe the behavior of any physics at the microscopic scale, not just atoms.
