How electric discharge happens in a discharge tube?

It depends on the type of tube. Hot cathode tubes use thermionic emission - that is, you heat a sample of an easily ionized substance (i.e., low work function) to a high temperature. A very small fraction of the atoms will be stripped away by the electric field."Cold cathode" discharge tubes use very high electric fields to strip the electrons. Again, a very small fraction of electrons will be stripped away from the cathode element.

The material at the cathode is the source of electrons. Only a very small fraction are ionized. Electrons eventually will be replenished from tiny leakage currents (the gas in the tube, the glass, etc.). The electrons at the anode initially will combine with positively charged ions (electron holes), which will then propogate as a slight current to the source of EMF.