We do not know why an accelerating body resists its acceleration. But if the worldtubes of physical bodies are real four-dimensional objects, the open questions of inertia and mass (as the measure of resistance a body offers to its acceleration) can be viewed from an unexpected point of view. As the worldtube of an accelerating body is deformed (is not geodesic) it seems natural to assume that there is a four-dimensional stress in the deformed worldtube of the body, which gives rise to a restoring force that resists the deformation of the body's worldtube and tries to restore its geodesic shape. This restoring force would manifest itself as the inertial force.
Calculations of the restoring force in the case of the classical electron and semiclassical calculations in quantum field theory show that the restoring force does have the form of the inertial force.