I bought some Whitney crab apples for pickling. I ate one prior, and it was fine. After I did the pickling, I cut a few open and there was a brown ring around the core of the seeds. Do you know what might have caused this?

The brown flesh around the core may have been caused by a fungus such as Alternaria sp.that entered through the calyx early in the fruit’s development. The condition, called moldy core, often cannot be detected until the fruit is harvested and cut. Here is a note about moldy core from the University of Massachusetts Apple Team:
“Moldy Core: This disease causes a rot with visible fungal growth in the seed cavity and core of some apple cultivars, particularly Delicious …

What is the method to get apple trees to set fruit?

All varieties of apple trees require some cross-pollination for fruit set. Even though some varieties are listed as self-fruitful, they will set fruit more heavily and more regularly if they are cross-pollinated. For solid block plantings, select varieties such as spur-type ‘Winter Banana’, spur-type ‘Golden Delicious’, or flowering crabapples such as ‘Manchurian’ and ‘Snodrift’ for pollinizers. Place honeybees in apple orchards when the first blossoms open. For more information on managing fruit trees, see this article on Pollination of Tree