The theory of evolution explains how strands of DNA change. An X-ray, cosmic ray, chemical reaction or similar mechanism can modify a base pair in the DNA strand to create a mutation, and this modification can lead to the creation of a new protein or enzyme.
The theory of evolution further proposes that billions of these mutations created all of the life forms we see today. An initial self-replicating molecule spontaneously formed. It evolved into single-cell organisms. These evolved into multi-cell organisms, which evolved into vertebrates like fish, and so on. In the process, DNA structures evolved from the asexual single-strand format found in bacteria today into the dual-strand chromosomal format found in all higher life forms. The number of chromosomes also proliferated. For example, fruit flies have five chromosomes, mice have 20, humans have 23 and dogs have 39.
Evolution's mutation mechanism does not explain how growth of a genome is possible. How can point mutations create new chromosomes or lengthen a strand of DNA? It is interesting to note that, in all of the selective breeding in dogs, there has been no change to the basic dog genome. All breeds of dog can still mate with one another. People have not seen any increase in dog's DNA, but have simply selected different genes from the existing dog gene pool to create the different breeds.
One line of research in this area focuses on transposons, or transposable elements, also referred to as "jumping genes." A transposon is a gene that is able to move or copy itself from one chromosome to another. The book "Molecular Biology of the Cell" puts it this way:
Transposable elements have also contributed to genome diversity in another way. When two transposable elements that are recognized by the same site-specific recombination enzyme (transposase) integrate into neighboring chromosomal sites, the DNA between them can become subject to transposition by the transposase. Because this provides a particularly effective pathway for the duplication and movement of exons (exon shuffling), these elements can help create new genes.
Another area of research involves polyploidy. Through the process of polyploidy, the total number of chromosomes can double, or a single chromosome can duplicate itself. This process is fairly common in plants, and explains why some plants can have as many as 100 chromosomes.
The amount of research in this area is truly remarkable and is teaching scientists amazing things about DNA. The following links give you a taste of that research, and are interesting if you would like to learn more about these topics: