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You add delta H fusion/vaporization because the enthalpy of the objects are different values when they are in different states, so you have to include that specific change in energy when calculating the total, if that makes sense! (for example, if you look at it in terms of water's heat curve, you can see the different enthalpy values for water in each state)
It takes heat to change the phase, so basically if you have ice and put it in water, it is first going to take the heat away from the water to melt the ice, and THEN the water will lose more heat to in turn heat up the "ice water." Therefore you add the heat required to melt and the heat to get to equilibrium temperature and set that equal to the heat lost of water.
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