What is the underlying reason why harmonics sound good?
The harmonic series consists of the fundamental, a frequency twice the fundamental, three times the fundamental, and so on. Doubling the frequency results in a note one octave higher than the fundamental. Tripling the frequency results in an octave and a fifth. Quadruple, two octaves. Quintuple, two octaves and a third. In terms of a piano keyboard you might start start with middle C, the first harmonic is the C above middle C, the G above that, the C two octaves above middle C, then the E above that.
The fundamental tone of any instrument usually sounds with a mixture of other frequencies. The piano string is free to vibrate along its full length, like a jump rope, or in halves, thirds, quarters. A single string sounds a series of notes in the harmonic series. Playing notes that match these notes produces a pleasant consonant sound. Notes which differ from those in the harmonic series produce other effects.
Understanding why the human ear finds these combinations to be pleasing is a much more complex question. The field of science called acoustics deals with a range of topics from how sound is produced, how it is transmitted through objects and the air, how the design of a room changes the way the sound behaves as it bounces off the walls, how sound is transmitted into the ear to become nerve signals to the brain, and finally the psychology of what meaning the human brain associates with those sounds.