Why are sine waves so fundamental?

Light waves, sound waves, ocean waves, spring motion and earthquake tremors, among many other natural phenomena, are all analyzed by being broken down into sums of sine waves. In each case, we are taking something that can be thought of as some signal strength that varies with time. Why is it so helpful to break things down into sine wave frequencies and why does this breakdown occur naturally? For example, we naturally break light down into different colors, as does a prism. We do a similar thing to sound waves. Why does the world seem to dance to the beat of sine waves?

I found this quote from a Wikipedia article.

The sine wave is important in physics because it retains its waveshape when added to another sine wave of the same frequency and arbitrary phase and magnitude. It is the only periodic waveform that has this property. This property leads to its importance in Fourier analysis and makes it acoustically unique.

This suggests two questions. Firstly, how the heck can you prove it and, for those who aren’t interested in the answer to that, why does this property make a difference?