Recombinant DNA technology involves isolating a specific fragment of DNA containing a desired gene, inserting this fragment into a plasmid or bacterial chromosome, and using the modified DNA to reproduce and synthesize copies of the gene. Key steps include extracting and purifying DNA, using restriction enzymes to fragment the DNA, isolating the fragment containing the target gene, ligating this fragment into a carrier molecule like a plasmid, transforming target cells with the recombinant DNA, and allowing the cells to replicate and express the inserted gene.