You probably already know about the "harmonic series" and overtones so I won't dwell on it here. Monarcheon can probably explain this better than I anyway.
But it's about overtones. If you play a fundamental of, say, C(3) on a piano, the following notes also sound: C(4), G(4), C(5), E(5) - etc. You don't hear them separately (unless you're listening out for them). They're what gives the piano and every other instrument its timbre (along with a few other parameters). Note that the fundamental produces a harmonic that turns up as the Major 3rd in a triad. C-E-G.
In a minor key, the 3rd is flattened. So in C minor: C Eb G.
Thus a discord occurs between the major 3rd in the instrument's natural sound and the minor 3rd.
It also explains why some instruments rich in harmonics sound angrier in a minor key than others. Compare a trio of horns (as in Peter and the Wolf for instance with the three flutes in the last movement of The Planets, The flute's lower register is weak in harmonics, the horns strong so the clash is greater.