I don't know enough about the mango specifically but, assuming it is like many other fruits, the change in taste is because complex carbohydrates (like starches) are being broken down to simple sugars during ripening. This will dramatically change the taste but will have little effect on the total calorie content since the total energy in the sugars will be close to (slightly less) that of the original starches.

As MikeAdams says, the starch is breaking down as the fruit ripens. This doesn't alter the calorific value of the fruit but it can change the amount of energy available to the human body because starch is inefficiently digested.

Someone might correct me on this, but the main digestion of starch into useable sugars begins in the mouth with the enzyme amylase. However, an acid environment (as present in fruit) suppresses this reaction. As far as I know there are no enzymes specific to starch digestion in the stomach or gut.