What is the logic behind this sentence in Keynes's General Theory?
This sentence says something very simple: If labor supply is a function of many arguments, not just the wage, $L^s = f(w,X,Y,...)$, then the supply curve, which is a line in a two-dimensional space of $(L^s,w)$ keeping everything else constant, will shift as a whole("bodily") if these $X's$ and $Y's$ change. 

What could these "$X$'s and $Y$'s" be? First, if the author defined labor supply as a function of nominal wages, then certainly, a change in prices of consumption goods would shift the labor supply curve.  

But even if the labor supply is defined as a function of real wages already, then there are other "$X$'s and $Y$'s" that may shift the whole curve, even if the real wage has not changed: for example, direct transfers from the Government (not as a windfall, but as a systematic policy). Since we assume that working is a "bad", people will want to work less for every real wage level, if they get an increased supplementary income through the Government.