Why do the units of rate constants change, and what does that physically mean?
Maybe it would make more sense if rate constants would have different symbols depending on whether it is a zero, first or second order reaction.
As an analogy, we might be asking &quot;how big&quot; an object is depending on the length of it. Depending whether the object is, say, a square, a cube or a hypercube, the answer would be in terms of its area, volume or 4D-volume, and the units (and dimensions) would be different in each case.
For the equilibrium constant, we avoid the issue by switching from concentrations to either concentrations divided by a standard concentration (this is a simplification) or to activities (the former multiplied by an activity coefficient that accounts for non-ideality).

Why do the dimensions of the rate constant change depending on different reaction orders, and what does that physically mean?

They have to change with the chosen system of units so that the equations are consistent. We could choose instead to have a dimensionless measure of concentration when we set up the kinetic equations. This would describe the reality equally well, but the kinetic constants would always have the same dimensions (1/time).
According to this argument, there is no physical meaning in rate constants having distinct dimensions depending on the order of reaction.