How is the mechanism for isolating DNA from an organism connected to the mechanism for genetically modifying that organism?

The genetic engineering process typically involves isolating DNA (genetic material) from one organism (an animal, plant, virus, or bacteria), and inserting it into the DNA of another organism, ensuring that the characteristics related with the inserted genes are conveyed in the modified organism. Crops that have been modified to express new traits via genetic engineering are also called "transgenic."