As the person proceeds up the steps, he changes his position and hence his potential energy. Each step of the staircase corresponds to a different energy level. Whe the person puts his foot on a step, he is at the energy level of that step. He cannot put his foot in between the steps, therefore, he cannot be at an "in-between" energy.

The person here is like the electrons. Likewise, the energy of the electrons can only have specific energy values associated with the energy levels.