This is how I reason it in my head.. without science and math.
Fundamental frequency is the frequency that the waveform repeats. If the waveform is a perfect sine, then there is only one frequency present.
If the signal is a square wave for example, then there must be higher frequencies present to create those "sharp edges".. and because the waveform repeats at the fundamental frequency, those higher frequencies need to be divisible by the fundamental frequency (synced to to the "sharp edges").