Your problem is apple maggots.  The only sure way to prevent the damage is to bag your apples when they are very small, about the size of a green olive.  There also are clay products that enclose the entire apple.  There also are sticky traps used in conjunction with a regular spraying schedule of pesticides.  Please open all the drop down boxes in this link: 

Apple maggots | UMN Extension

Thank you. I have been using Neem oil. I spray the trees after the buds fall off and then about 2 to 3 weeks later and then around July 4. Is neem oil effective? Should I just continue it every 2 to 3 weeks after July 4. Our trees are quite big and bagging all the apples would be quite tedious.

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Neem oil is not the recommended product. You should be using Fruit Tree Spray and following the directions on the label which is probably every two weeks, after a rain, etc.  You can still bag the apples on the lower branches.  Bag just as many at that level that you want to save and spray the others.