Why enthalpy changes in phase transition when there is no change of temperature?
For a single phase, enthalpy is a function of temperature and pressure, and, at constant pressure, $\mathrm dH = C_p\,\mathrm dT.$ 

However, at a change of phase, enthalpy (per unit mass) undergoes a jump change, even at constant temperature and pressure.  If the phase change is from liquid to vapor, for example, this jump change in enthalpy is called the heat of vaporization.