This is a little tricky, actually. You could say that the reason everything is moving away from everything else is the expansion of space. But you could also argue the reason we say space is expanding is that everything is moving away from everything else - after all, what does it mean for space to expand, other than that the distances between objects get larger?
We do have a perfectly clear mathematical description of cosmic expansion. It centers on the observational fact that the distance between any two objects at large enough scales (galaxy clusters and superclusters, not atoms or stars) is described by the equation
$$r(t_1) = \frac{a(t_1)}{a(t_2)}r(t_2)$$
The scale factor $a(\tau)$ changes in time in a way determined by cosmological equations, basically general relativity with some particular metric. (The FLRW metric is an artificial example.) This allows us to unambiguously determine how the distance between any two objects changes over time. But whether you interpret that change in distance as being a consequence of the expansion of space, or of space being created (though I really dislike that interpretation), or just actually being the expansion of space itself, is to some extent a matter of terminology.
Inertia is definitely not responsible, though. Inertia is a fine concept to apply when you're talking about small-scale objects, where you can ignore the effects of gravity; this allows you to have locally inertial reference frames in general relativity. But there are no globally inertial reference frames; informally, you can't continuously extend the concept of inertia on Earth's surface to, say, the surface of a neutron star, and certainly not throughout the universe.
Further reading: Why space expansion affects matter? and many other questions on cosmology on this site.