If a reaction is zeroth order overall, it means that the rate of the reaction is independent of all of the concentrations of the reactant. This means that the rate of this reaction will not change as the reaction proceeds. Typically as a reaction progresses, the concentrations of the reactants decreases and the rate decreases. However, for zeroth order reactions, this is not true, the rate is constant in time.
This is because the rate law is
\[\rm{rate = k (constant)}\]
Assuming we are looking at some generic reactant A we can now write
\[\rm{{-d[A] \over dt} = rate = k}\]
this can be easily integrated up to yield
\[\rm{[A] = [A]_0 - kt}\]
The concentration of A is linearly decreasing in time from the initial concentration.
Thus for a zeroth order reaction, a plot of the concentration as a function of time should yield a linear plot.
Zeroth Order Integrated Rate Law