Doubt about Van Der Waals equation
In the ideal gas equation $$p_iV_i = nRT$$ $p_i$ is the pressure if there was no intermolecular forces between the molecules and $V_i$ is the free volume available for the molecules to do motion in, which is the volume of the container.
In real gases however, the pressure that you observe (real pressure) is less than the expected pressure. This is because of the intermolecular forces. So you convert the $p_r$ to a pressure which the gas would exert if there were no intermolecular forces.  $$p + a\frac{n^2}{V^2}$$
The free volume available to real gas is less than the volume considered in ideal gas equation. That is because Kinetic theory of gases made an assumption that the volume occupied by molecules of an ideal gas is negligible. But in real gases, the molecules are of considerable size. Thus, the full container will not be free for motion, some volume of the container will be occupied by the molecules. Let the volume occupied by one mole gas molecules be $b$, for n mole gas this is $nb$, so free volume available to gas is less than the volume of container $$V_i =  V_{\text container} - nb$$
Edit: In your question, the $V_r$ they're talking about is volume of the container. And $V_i$ is actual free volume available to the real gas.

Real Volume: Volume of the container
Ideal volume: Free space available
Real pressure: Observed pressure
Ideal pressure: Pressure if the gas didn't have any intermolecular forces.