They arise naturally in Fourier analysis because we are trying to express functions (sometimes pure real functions) as sums of complex exponential. There is no way to make a real function out of complex exponential unless we add two complex conjugates. exp(j*w*t) and exp(j*<-w>*t) are complex conjugates.
For examlple, try to make cos(w0*t) using functions exp(j*w*t).
It can be made by two components at w=w0 (positive frequency) and w=-w0 (negative frequency)
cos(w0*t) = 0.5*[exp(j*w0*t) + exp(j*<-w0>*t)]