Looking at the articles referenced in the Wikipedia article, there's probably no direct physiological link between senescence and beta-galactosides. Lee et al (2006) and others before them have shown that the "hypothetical" protein is just regular lysosomal beta-galactosidase, which is present in higher concentrations in aging and stressed cells because the number of lysosomes increase under those conditions. There's a collection of references in the Discussion section of the Lee paper that delves deeper into the aging-lysosome connection.
I'm well out of my depth by this point, but if I were going to take a wild guess I'd suggest that lysosomal upregulation could be part of a stress response involving autophagy (i.e. the cell breaking down and recycling intracellular materials to meet critical demands), though I don't know the specific connection to aging. As far as beta-galactosides go: if you have any hanging around, you get to eat them, is all, I bet. I bet the references in Lee would tell you if I'm on track or not.