Classically emission is continuous and the electron would need to occupy a "in between" energy level for a while, and that is forbidden in Bohr's scheme, so the emission can't be allowed to happen.
This doesn't really explain why it can't happen, but that's phenomenology for you: you keep lining up facts until your kludge (1) gets the right answer and (2) might be pointing at a better "real" theory.
This is the subject of an underrated classic paper from the early days of quantum mechanics:
Mott's introduction is better than my attempt to paraphrase:
In the theory of radioactive disintegration, as presented by Gamow, the $\alpha$-particle is represented by a spherical wave which slowly leaks out of the nucleus. On the other hand, the $\alpha$-particle, once emerged, has particle-like properties, the most striking being the ray tracks that it forms in a Wilson cloud chamber. It is a little difficult to picture how it is that an outgoing spherical wave can produce a straight track; we think intuitively that it should ionise atoms at random throughout space. We could consider that Gamow’s outgoing spherical wave should give the probability of disintegration, but that, when the particle is outside the nucleus, it should be represented by a wave packet moving in a definite direction, so as to produce a straight track. But it ought not to be necessary to do this. The wave mechanics unaided ought to be able to predict the possible results of any observation that we could make on a system, without invoking, until the moment at which the observation is made, the classical particle-like properties of the electrons or $\alpha$-particles forming that system.
Mott's solution is to consider the alpha particle and the first two atoms which it ionizes as a single quantum-mechanical system with three parts, with the result
We shall then show that the atoms cannot both be ionised unless they lie in a straight line with the radioactive nucleus.
That is to say, your question gets the situation backwards. The issue isn't that "free electrons have classical trajectories," and that these electrons are "not able to move classically anymore" when they are bound. Mott's paper shows that the wave mechanics, which successfully predicts the behavior of bound electrons, also predicts the emergence of straight-line ionization trajectories.
With modern buzzwords, we might say that the "classical trajectory" is an "emergent phenomenon" due to the "entanglement" of the alpha particle with the quantum-mechanical constituents of the detector. But this classic paper predates all of those buzzwords and is better without them. The observation is that the probabilities of successive ionization events are correlated, and that the correlation works out to depend on the geometry of the "track" in a way which satisfies our classical intuition.
Best Answer
You are right, the planetary model of the atom does not make sense when one considers the electromagnetic forces involved. The electron in an orbit is accelerating continuously and would thus radiate away its energy and fall into the nucleus.
One of the reasons for "inventing" quantum mechanics was exactly this conundrum.
The Bohr model was proposed to solve this, by stipulating that the orbits were closed and quantized and no energy could be lost while the electron was in orbit, thus creating the stability of the atom necessary to form solids and liquids. It also explained the lines observed in the spectra from excited atoms as transitions between orbits.
If you study further into physics you will learn about quantum mechanics and the axioms and postulates that form the equations whose solutions give exact numbers for what was the first guess at a model of the atom.
Quantum mechanics is accepted as the underlying level of all physical forces at the microscopic level, and sometimes quantum mechanics can be seen macroscopically, as with superconductivity, for example. Macroscopic forces, like those due to classical electric and magnetic fields, are limiting cases of the real forces which reign microscopically.