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Unlike a lot of things we see in daily life (ex: a continuous stream of water), electrons only exist on specific energy levels (n=1, n=2, etc) and nothing else. This means that they can't be found anywhere in between these values. Being quantized means that the amount of energy absorbed or released can only exist in discrete amounts. So when electrons are changing energy levels, they can only absorb or emit the exact amount of energy needed to take them to another level.
All it means to be quantized is that the electrons can only have specific values for their energies; in other words, its values are discrete and not continuous. For example, its energy can only be 1,2,3,4... (again just as an example to illustrate the point) and cannot be 1.387 or 4.821 because they are not discrete values they're continuous. The idea of electrons being quantized is part of the reason why the Bohr model of atoms is still taught in schools because even though it is an oversimplified model of an atom, it was an important development because it established the idea that electrons are quantized.
At the quantum level, the an electron can only occupy a certain energy level. This means that a specific amount of energy, i.e. a quantized amount is required to raise an electrons energy. This energy differs across the elements.
When he says this, he means that the electron can occupy only specific levels of energy (n=1,2,3,4...). The electron can never be in between such levels of energy, but rather, must occupy a discrete quantum level of energy. When an electron loses energy, it drops to a lower quantum level and the lost energy is given off as a photon.
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