"1st, 3rd and 5th degrees of the scale are relatively stable"
That cannot refer to chords but individual notes or pitches, when there's an established tonic pitch in the listener's mind. The major chord built on the 5th scale degree is definitely not "stable" relative to the tonic, because that's the dominant chord. If your tonic is C, in C major, even if it's only G major and not G7 (which would be the proper "dominant seventh" chord for C), it makes you want to hear a C as the next chord to make it feel like the harmony resolved.
So it's a misunderstanding. The somewhere you're quoting is talking about individual pitches when there's a tonic, which is the first scale degree. "Stability" means lack of sense of pressure to change. If you have established C as the tonic, and you play an F - the 4th scale degree, or a B - the 7th scale degree, it makes you feel tense: things are not right until you hear a C, E or G note. That's the meaning of "unstable". If you hear a G note, which is the 5th scale degree, you feel like, "that's not perhaps an ideal final solution, but it's ok, let it be at that". That's "stable".
All of this depends on there being an established tonic note.
What comes to the physics side and "consonance" and "dissonance" and all ... how do you explain that the fourth scale degree is "very unstable", when the interval between the root and the fourth is a perfect interval and a perfect consonance? http://www2.siba.fi/muste1/index.php?id=65&la=en
So you "resolve" the tension caused by the fourth scale degree by moving it to the third scale degree, how come? The interval between the root and the third is said to be more dissonant than the fourth, so you resolve an instability by increasing "dissonance"? It must be slightly more complicated than a simple physical measurement between frequencies.
I guess a proper answer to this question would need to define all the terms: pitch, scale, scale degree, interval, root, chord, stability, instability, tension, resolution, dissonance, consonance, ... hopefully someone writes a better answer that sets everything straight. :)
However, I wouldn't put too much weight on these explanations, because their actual real-life utility is limited. Music theory describes feelings and phenomena occurring in practice, and the descriptions can be helpful in finding your way around when composing or playing music, or when talking and reasoning about it. But if you sit at a piano and play these notes, scales, intervals and chords, you can experience first hand how the things affect your feelings, without needing a degree on theoretical concepts.
Anyway, the title of this question is a misunderstanding. The quoted text is not talking about stability of chords.
Here is some further reading about stability from this site:
I'm still not sure how you could utilize these "why" explanations. Would any of them affect your music making? Maybe they can help you to see patterns and relate things to them.