Suppose I have the utility function $$U(x,y) = \sqrt{x} + y$$ subject to budget constraint $$p_x x + p_y y = m$$ Then $$x_M =\frac{p_y^2}{4 p_x^2}$$ $$y_M = \frac{m}{p_y} - \frac{p_y}{4 p_x}$$
where $M$ denotes Marshallian.
Now suppose I increase $p_x$ to $p_x'$.
Why is the income effect zero?
From the formula for $x_M$, we see it has no dependence on income $m$. So $$\frac{\partial x_M}{\partial m} =0$$ Thus, the Slutsky equation
$$\frac{\partial x_M}{\partial p_x} = \frac{\partial x_H}{\partial p_x} +-\frac{\partial x_M}{\partial m}x_M$$
implies
$$\frac{\partial x_M}{\partial p_x} = \frac{\partial x_H}{\partial p_x} +(0)x_M $$ $$\frac{\partial x_M}{\partial p_x} = \frac{\partial x_H}{\partial p_x} $$ Hence, and since $-\frac{\partial x_M}{\partial m}x_M$ is the income effect, this implies the income effect is zero and all the change is due to the substitution effect.
Intuitively, the marginal utility of x falls faster than the marginal utility of y (which is actually constant), so with enough money all marginal funds go into y. Similarly, with enough money, an decrease in money only reduces the quantity of y, not x. But you should know this isn't true globally. If $$m < \frac{p_y^2}{4 p_x}$$ then the demand functions are: $$x_M =\frac{m}{p_x}$$ $$y_M = 0$$
