To put in "Frequency" terms, the value of zero frequency corresponds to "DC" by name and it has nothing to do with Direct current actually. The fourier transform at zero frequency is simply an integration of a signal throughout its existence (dividing by the length of the time interval gives the mean value of the signal). Now if the Integration(in the case of continuous time domain) or the summation(discrete time domain) turns out to be zero, essentially you won't have any zero frequency component, and hence the average will also be zero.
Also you have to be careful with the fact that the transforms exist only if the signals don't blow up. Eg. a signal that's just a constant from $-\infty$ to $+\infty$ will blow up at zero frequency. In real world we deal with time limited signals and the signals that don't blow up so it's not an issue.