Science never answers "why" questions, so in a strict sense there is no such explanation, but one can try to triangulate where we stand, at the moment.
In classical physics space, time and the existence of massive bodies are inexplicable pre-physical facts. Inertia then becomes an observed property of massive bodies that allows to differentiate them by their mass. Together with the observed homogeneity of space and time and the isotropy of space much of classical physics follows from there.
Relativity unifies space/time and energy/momentum (including rest mass) and inertia becomes a necessary property of systems with localized binding energy, but relativity alone can't describe the internal details of such systems. For that purpose we need quantum mechanics, more precisely quantum field theory. At this level we get to learn that the symmetry properties of spacetime have very profound consequences for matter and its interactions, but we are still short of having found a self-consistent explanation for spacetime itself. So one could say that among the classical mechanics triad space/time/matter, which is needed to "have a home for inertia", we have succeeded somewhat in understanding matter to some extent and we have (somewhat) unified space and time, but the fundamental object of "spacetime", which gives rise to all of this, is still not understood.
For me, and this is an opinion, of course, the fundamental question of "How does inertia really work?" (which goes beyond "What does inertia do?"), can not be successfully answered until we have made successful inroad into the "How does spacetime really work?" question. I do not expect rapid experimental and observational progress on that front because spacetime has turned out to be an extremely smooth object, so far... and we just don't have the right microscopes, yet, to see the structures in it that will get us towards a microscopic understanding.
Does one need inertia for equilibrium? No, but without it there can't be anything else. In classical physics inertia sets the velocity scale and without it dynamic equations are meaningless. In relativity all systems without rest-mass are moving at the same velocity in all observer systems, which is not very interesting. What makes the world so rich is the interplay between massive and massless fields. I believe the theoreticians can explain why one can't exist without the other, but I don't think that answers your question, either.