There are a variety of ways to genetically modify an organism. It's not even that difficult depending on the method. Genetically modifying organisms has been happening for centuries.
The easiest, or lowest tech method, is selective breeding. Every organism has different traits that are determined by the set of genes that it inherits. A breeder can choose to mate organism "A" with organism "B" if they each have desired traits. Or "A" and "B" may be mixed in order to produce offspring that have two different desired traits.
A different way to create a GMO is to use gene therapy. This is a huge catch all name for anytime a geneticist intentionally modifies a specific sequence of genetic coding. It's possible to insert the actual DNA sequence into the nucleus of the organisms cells, but it requires a crazy small syringe to do. A "simpler" method is to use a virus. Remove the viral DNA, replace it with the desired genetic code, and inject the virus into the organism. The virus will "infect" cells with the desired genetic information. This was first done on a girl named Ashanti Desilva who suffered from a rare immune disease called ADA. I've attached a link.