Why is there a subtractive volume correction in the van der Waals' equation for real gases?
The van der Waals equation is an attempt to explain the pressure-volume relationship for real gases. For example, what will the pressure be of a given amount of gas in a vessel of a fixed volume.

In ideal gases the size and interactions between the atoms or molecules of the gas don't matter at all. We think of them as point-like entities that don't interact except to bounce elastically off each other and off the walls of the vessel. Their behaviour is described by the ideal gas equation and doesn't depend at all on the nature of the particles making up the gas.

Real molecules and atoms deviate from this ideal behaviour because the particles of the gas both have finite volume and attractive interactions which mean they don't behave exactly as the ideal gas equation describes. The van der Waals equation is a simple attempt to explain those deviations from ideality. The volume correction is one of those factors. It takes into account the fact that real gases are made from particles that actually have a finite size and are not mathematical points. The factor is about the volume of the particles of the gas not about the vessel the gas is in. The result of the correction is that the pressure of a given amount of gas in a vessel of fixed size is slightly different than would be expected from the ideal gas equation.