I know that the sig~ object converts numbers into an audio signal, but I would like to know what the object is doing in more detail. Would the input of the object be the frequency of the signal? the amplitude? the phase? What aspect of the signal would the input represent.

Would the output of a sig~ object be a sinusoidal signal? would it have a random waveform? or what kind of signal would the output be??

it just takes the last number/ first argument and outputs that number as a constant audio signal every block. every block has constant values I believe. the input/output is a float, whatever you want it to be. If you plug it into an oscillator it will be the frequency, if you use it as input to a transfer function it could be phase.
whatever it is, i think it only changes every block (usually 64 samples)