Where does the sum current of the three currents flow?
if each individual current enters via Phases R, Y and B, where does
  the sum of the three currents (that is Ia+Ib+Ic) flow after meeting at
  the common point of the three phases?


The "sum" current has to be zero because it can't flow anywhere.


  if I were to keep an ammeter, I can individually measure the value of
  the three currents right? So how exactly does it become zero?


Your ammeter doesn't account for the fact that the three currents have a phase relationship that means they cancel out. Individually they can be measured but I_red = -(I_yellow + I_blue)

Or, I_red + I_yellow + I_blue = 0