When we hear a note it is made up of a fundamental frequency and many partials, which help us recognise the timbre as that of a piano, or a clarinet. The harmonic components of the sound can be particularly strongly heard in, say, a distorted guitar signal. But any fifth you can hear will be slightly sharper than a fifth derived from equal temperament (e.g. 7 frets' difference on a guitar string).
If you mean, why is tonal music structured around fifth relationships (dominants resolving to tonics etc.), then the answer is debatable. But the reason I pointed out the tuning distinction above is because one might be inclined to say that tonal music (which includes most pop as well as classical) was derived from the 'natural' laws of Pythagorean intervals. This would make the cycle-of-fifths a 'natural' phenomenon. But actually tonality relies on a compromised tuning system, so this doesn't quite explain the success of tonal music. It's worth noting that tonality is really a Western thing, and so we don't 'naturally' hear tonality so much as we've been 'nurtured' to hear it -- from the moment we hear the radio or TV, or whatever.
As to why we compose using fifth relationships -- an equally tempered system means that, following your chain of 'F, C, G, D, A, E, B' if we continue this into chromatic notes, eventually we reach the first note again. So fully-fledged tonality offers the most flexible system for composers to modulate to any key they like without the worry of the tuning not working. In many alternative tuning systems, we can only stray so far by fifths, until the key we reach isn't tuned in a way that pleases our ears! However, the trade-off is that, other tuning systems can offer 'nicer' sounding fifths and thirds, if you stick to the home key. Interestingly, a choir or a string quartet isn't constrained by say the frets on a guitar, so they can adjust the tuning of their perfect intervals regardless of which key they're in.
Hope this helps!