You can look the gene up on JAX informatics for a more detailed description. All I know is the mouse's obesity is directly linked with the amount of yellow pigment in the coat (even if the mouse is diluted by c-dilutes, it has to do with the area of the mouse that would be white). The more stripes you have on a brindle the less likely they are to become obese.
Think of it like Dominant Yellow (Red) mice who are A^y, their obesity is linked to the actual gene, and they are fully yellow, the Viable Yellow gene, A^vy, inhibits some of the yellow pigment and causes brindle stripes, but it is the same as Dominant Yellow, essentially.
It is called Viable Yellow because the homozygotes are viable, unlike the homozygotes in Dominant Yellow, which die before implantation or shortly thereafter.
I knew all that about the yellow pigmentation, but I want to know why a color gene is linked to a metabolic issue (chemically) .. if that's what's going on. I'll go look it up on the JAX site though - thank you!