why do these energy levels mean that the atom will show an emission spectrum of individual lines rather than a continuous spectrum of emitted lights?

It is a fundamental concept that atomic spectra are discontinuous the electron's energy levels are quantized which is a quantum mechanics concept that a system is limited to certain energy states which depends on the particular system being refereed to

Explanation:

Atomic electrons posses only specific energy levels with nothing in between those levels, it can be excited to a higher energy level by gaining the required energy or lose energy in quantized packets to drop to a lower energy level hence the discontinuous spectral lines