This quote is referring to the use of genetic engineering to allow parents to select traits for their children. The geneticist is referring to the fact that each parent gives 23 chromosomes to their child resulting in 23 homologus pairs of chromosomes. The homologus pair consists of matching chromosomes, one from each parent. The idea behind this quote is that the doctors select the best possible chromosomes from the mother and match them with the best possible homologus chromosome from the father.
The chromosomes are composed of a series of genes which control the traits that are expressed. The alleles are the possible forms of a trait. For example, for dimples, there is a dominant allele and a recessive allele. If you receive at least one dominant allele then you will express the trait and have dimples, if you receive two recessive alleles (one from each parent) then you will not express the trait and you will not have dimples.
In this situation the parents basically have the opportunity to select which alleles they would like their offspring to have. Essentially the child will have all of their parents DNA however it will be the best possible combination of alleles resulting in the best possible offspring.