The fundamental reason we define intervals as pairs of pitches with a certain ratio (e.g., 3/2), rather than a certain fixed frequency difference (e.g, 100Hz), is that pitched instruments create overtones. In a simplified picture, if you pluck a guitar string, you will get both the fundamental frequency (let's say, 220 Hz), which corresponds the entire string oscillating back and forth. You will also get and multiples of it: twice the base frequency (440 Hz), which corresponds to two halves of the string oscillating in opposite directions, and the point in the middle staying at rest; three times the fundamental frequency (660 Hz), where the string is divided into three parts oscillating in opposite directions to their neighbors, etc.
Similar things happen with the oscillating air column in wind instruments.
The absolute frequency does not matter here (within the range of the instrument) - a guitar string sounds (pretty much) the same whether you play it at 220 Hz or 180 or 270, because the frequency ratios and amplitude ratios of the overtones to the fundamental frequency are the same.
So the sound of each pitched instrument is a combination of all these frequencies, or overtones. If you combine two of these pitches, their overtones also add up - some are the same frequency, so they reinforce each other; some have frequencies in complex ratios, which tend to sound dissonant. Again, how the overtones add up is independent of the absolute frequency - you get pretty much the same sound impression if you play notes at 220 and 330 Hz, or at 300 and 450, because they're at the same ratio of 3/2, and their overtones stack up the same way.
The rest of western music theory (which notes to use together in scales, how to combine them into chords, which chords sound consonant or dissonant...) is basically a very elaborate heuristic (rules based mostly on experience) on how the combinations of overtones combine and how these combinations are perceived by the listener, mixed with habits developed during centuries of a certain musical tradition and the constraints of the instruments commonly used in that tradition.