If you look at the spectrum below of light emitted by energetic atoms (such as the orange-yellow light from sodium vapor street lights, or the blue-white light from mercury vapor lamps) you will notice that it is composed of individual lines of different colors. These lines represent the discrete energy levels of the electrons in those excited atoms. When an electron in a high energy state jumps down to a lower one, the atom emits a photon of light which corresponds to the exact energy difference of those two levels (conservation of energy). The bigger the energy difference, the more energetic the photon will be, and the closer its color will be to the violet end of the spectrum. If electrons were not restricted to discrete energy levels, the spectrum from an excited atom would be a continuous spread of colors from red to violet with no individual lines.
Eventually all this early work by these men from the 19th century and later the work of Irwin Shroedinger from the 20th century led to today's modern understanding of Quantum Mechanics, which is for a later topic. In Conclusion, it is the science of spectrosopy that tell's us what elements the stars are made of.