There are reasons grounded in the physics of soundwaves for why frequencies related by simple ratios sound "special" when played together. As geofft explained, they naturally arise in any physical instrument due to how things like strings (guitar, piano, etc.) or air in a tube (wind instruments, etc.) vibrate. Another way to look at this is to think of two waves with wavelenghts that are related in say a 3/2 ratio (wavelength of a wave is essentially just 1 over the frequency of the wave, so you can think in terms of either, but wavelengths are easier to visualise). If you combine two sine waves with such wavelenghts by summing them up, they form a very clear repeating structure, where the crests and valleys of the sum-wave have a regular pattern to them that repeats: https://www.wolframalpha.com/input/?i=plot+sin%28x%29+%2B+si... Whereas if the wavelenghts of the two waves being combined are not related to each other by a simple ratio, the crests and valleys of the sum-wave keep moving and changing instead of repeating: https://www.wolframalpha.com/input/?i=plot+sin%28x%29+%2B+si... When heard as sound, the fixed repeating structure of the combined wave when the two component waves "line up" with each other has a special quality to it, that we would call harmonious or consonant.
So in the above sense, and because of overtones as explained by geofft, just intonation really is special among various tuning systems. Of course there's still the separate question of if the difference between just intonation and 12-EDO (what most music uses) is big enough for the average listener to notice. This you can test for yourself by playing on Jintone e.g. a just intonation major chord, for instance 1/1, 5/4, and 3/2 together, and then the same chord in 12-EDO by clicking the piano keys directly below the tones you just played, so C, E, and G. The difference isn't like night and day, but the latter pretty clearly has an unstable wobbliness in it that's not there in the just intonation version, a difference that I think most people would detect. If you want to make the difference super obvious, click the gear icon on top right and choose the sawtooth waveform, and repeat the exercise. The timbre of the sawtooth waveform makes the difference more obvious for exactly the reason geofft was talking about: It's a waveform that has a lot of very strong overtones in it (kinda the opposite of a pure sine wave), and in the just intonation chord the overtones of the three tones in the chord match exactly.