What is the origin of such a wide genetic variation among offspring?

In all the species that reproduce sexually, there are three main mechanisms that contribute to genetic variation:independent assortment of chromosomes, crossing-over, and random fertilization. Independent assortment of chromosomes means that during meiosis 1 each homologous pair, consisting of 2 chromosomes, can orient either with maternal or paternal chromosome to a given pole of the cell. Therefore, each daughter cell can get a huge number of different combinations of maternal and paternal chromosomes.Crossing over, process that also happens during meiosis 1, is gene exchange between homologous chromosomes. It produces chromosomes with new combinations of genes from maternal and paternal chromosomes - recombinant chromosomes. Finally, random fertilization means that male gamete will fuse with a random female gamete which increases number of combinations by one order of magnitude.