They suggest that altered brain activity patterns are what contribute to this phenotype: something that I wouldn't have guessed at first. I would have thought that there was something about their skeletal or muscular morphology that contributes to the ability or desire to do this (like, it just feels more comfortable to walk that way). But apparently these people have cerebellar hypoplasia.
If you look at the paper, there are some references to videos of these people.
Surprisingly the genetic basis for this trait is heterogeneous. Not all of them have the same genetic association.
from the conclusion:
In conclusion, we suggest that VLDLR-deficiency in the brain at a key stage of development leads to abnormal formation of the neural structures that are critical for gait. Given the heterogeneity of causes of quadrupedal gait, identification of the genes in families B and C promises to offer insights into neurodevelopmental mechanisms mediating gait in humans.Mutations in the very low-density lipoprotein receptor VLDLR cause cerebellar hypoplasia and quadrupedal locomotion in humans
Tayfun Ozcelik, Nurten Akarsu, Elif Uz, Safak Caglayan, Suleyman Gulsuner, Onur Emre Onat, Meliha Tan, and Uner Tan
PNAS Published online on March 7, 2008
Abstract Quadrupedal gait in humans, also known as Unertan syndrome, is a rare phenotype associated with dysarthric speech, mental retardation, and varying degrees of cerebrocerebellar hypoplasia. Four large consanguineous kindreds from Turkey manifest this phenotype. In two families (A and D), shared homozygosity among affected relatives mapped the trait to a 1.3-Mb region of chromosome 9p24. This genomic region includes the VLDLR gene, which encodes the very low-density lipoprotein receptor, a component of the reelin signaling pathway involved in neuroblast migration in the cerebral cortex and cerebellum. Sequence analysis of VLDLR revealed nonsense mutation R257X in family A and single-nucleotide deletion c2339delT in family D. Both these mutations are predicted to lead to truncated proteins lacking transmembrane and signaling domains. In two other families (B and C), the phenotype is not linked to chromosome 9p. Our data indicate that mutations in VLDLR impair cerebrocerebellar function, conferring in these families a dramatic influence on gait, and that hereditary disorders associated with quadrupedal gait in humans are genetically heterogeneous.