Negative frequencies are not intuitive. They appear in Fourier transforms because those transforms present signals as sums of complex exponentials exp(j(2Pi)ft).
Presenting signals as sums of usual sines would be perfectly possible and they would not need negative frequencies, but algebraic manipulations and formula derivations would become much more difficult.
In math we could use any orthogonal base function set, so complex exponentials are in that sense not better nor worse than the common real sine functions. But complex exponentials are easier to manipulate and the connection between them and real sinusoidals is simple - the spectrum is mirrored at negative frequencies.
So: For pure convenience we have taken into use negative frequencies and calculate spectrums assuming the signal is a sum of complex exponentials. The price is that every spectrum of normal time dependent voltages must be mirrored to the negative frequency side and every practical filter frequency response must be mirrored in the same way. That's no problem after one has got some time to adapt himself to the convention.
There's more advantages than easier algebraic manipulations. Some advanced signal processing will become possible if we allow signals to have also the imaginary part. That's, of course only a separate memory word in digital signal processing. Real part is in one word and the imaginary part is in another. Spectrums of those signals are not symmetric around f=0, but they are calculated with the same Fourier transform formulas with complex exponentials.
You may see in the principle explanations of modulators and detectors I and Q channels. Those applications are perfect examples of the advantages of complex valued signals.
Let's assume you have built an amp or filter circuit which handles practical voltage signals say 0 to 1MHz. You can in normal talk say that it covers band 0...1MHz, but when calculating exactly its effect to signal spectrums which have negative frequencies you should automatically keep in mind that in frequency domain it affects symmetrically around f=0.