What is the physical significance of negative frequencies?
Negative frequency doesn't make much sense for sinusoids, but the Fourier transform doesn't break up a signal into sinusoids, it breaks it up into complex exponentials (also called "complex sinusoids" or "cisoids"):

$$F(\omega) = \int_{-\infty}^{\infty} f(t) \color{Red}{e^{- j\omega t}}\,dt$$

These are actually spirals, spinning around in the complex plane:



(Source: Richard Lyons)

Spirals can be either left-handed or right-handed (rotating clockwise or counterclockwise), which is where the concept of negative frequency comes from.  You can also think of it as the phase angle going forward or backward in time.

In the case of real signals, there are always two equal-amplitude complex exponentials, rotating in opposite directions, so that their real parts combine and imaginary parts cancel out, leaving only a real sinusoid as the result.  This is why the spectrum of a sine wave always has 2 spikes, one positive frequency and one negative.  Depending on the phase of the two spirals, they could cancel out, leaving a purely real sine wave, or a real cosine wave, or a purely imaginary sine wave, etc.

The negative and positive frequency components are both necessary to produce the real signal, but if you already know that it's a real signal, the other side of the spectrum doesn't provide any extra information, so it's often hand-waved and ignored.  For the general case of complex signals, you need to know both sides of the frequency spectrum.