How to explain the cosmic expansion?
I'm going to explicate Conifold's reference a bit because while he provides the theory that claims to explain (which is physics proper), he doesn't address the how or the why of the explanation (which is metaphysics proper). The TLDR of the question of 'how to explain' is using instrumentalist reasoning consistent with the edifice of the contemporary paradigms of mathematical physics. Let's explore the current scientific methodology for generating scientific explanation of this sort.
is the question ill-posed?
Absolutely not, because the heart of science is providing explanation, but explanation like causality is a metaphysically challenging topic. Just as one can reject causality as a linguistic fiction of highly probable correlation (See Judea Pearl's work on Causality for a detailed mathematical treatment of causality), so too it begs the question 'is an explanation real in any sense beyond it can be experienced and if so how and why'.
Do science or natural philosophy make any proposal concerning the driving force of the expansion?
Conifold has directed you to dark energy and dark matter conceptually. But is dark matter explanatory and if so, in what way? Does it fit Hempel's deductive-nomological structure? Are there statistical formulations of it as in causality? Is the primary method to explain 'expansion' by causal claims? This is where the philosophy of science has to earn its paycheck.
It's easy to point to 'dark matter' as an explanation for the dynamics of 'expansion of the universe', but all sorts of philosophical questions arise. But is such an explanation scientific, and if so, how do we know? How can a universe expand if by definition there is nothing beyond the universe to expand into? How can dark matter be matter at all if it can't be detected? And how does something that seems to have no physical existence directly or indirectly observable explain something that linguistically can't happen. Enter in mathematical physics.
Since the quantum revolution, what has become apparent is that the explanatory function of science has taken on two forms, one that is realist and one that is instrumentalist. Therefore since cosmic expansion is a mathematical model of the universe, to explain it is to explain the mathematical concepts that drive the 'expansion'. A blueberry muffin in an oven is but a model to simplify understanding the mathematics of expansion by appeal to intuition.
Newton's law of gravitation is at the heart of explanation of the effects of gravity, and the relative acceleration of the rate of velocity in the expansion of celestial bodies relies on the fact that acceleration is defined in terms of both mass and force. So, if the universe is expanding and doing so with acceleration and gravity, there can only be one cause of this: mass. Yet, that's problematic because no such mass seems to have been observed to cause the acceleration. Therefore, physicists find themselves in a bit of a contradictory situation. On the one hand, only mass can explain the acceleration, and yet there is no observation of this extra mass to do so. Something has to give.
Physicists simply become philosophers and embrace a new ontological commitment. They engage in semantic ascent and perform existential quantification. They declare 'there is extra mass' (which is energy which is matter by the mass-matter relationship and the mass-energy equivalence Einstein put forth) and label it "dark" and unobservable. Then they get busy trying to use the mathematical model of the standard particle model to try figure out what it means mathematically to be dark matter or dark energy. The details of which are physics and not philosophy.
So, you asked:
How to explain the cosmic expansion?
To a philosopher of science, then, what is of interest here is not so much the answer physicists have cooked up, the dark matter and energy (which are arguably empirical only through a convoluted notion of empirical), but the methodology. Thus, how to explain cosmic expansion is to define it mathematically to align with observations, reason about the causes of the acceleration of bodies using Newtonian laws and GR, and then create a new entity de novo, an alternative to classical notions of matter and energy. This is the actual how of scientific explanation in contemporary mathematical physics that astrophysicists use, and therefore can be shown to be consistent with the various scientific methodologies and methodological naturalism philosophically.