I will try to answer what he may have meant, or what an arbitrary person saying that may have meant. I did go read the article.
The bottom line is that there is not much content in the separation he made between a spacetime theory of gravity and a metric field theory of gravity. It is more than anything just a bit of a philosophical difference, and a bit a way of thinking, whether as a dynamic geometry or as dynamic fields in a fixed geometry. But since the results are the same, as he admits, it makes no physical difference.
In Living Reviews in Relativity at http://relativity.livingreviews.org/Articles/lrr-2014-4/articlese3.html, they review the status of different theories of gravity. Interestingly, for metric theories, there is always a spacetime. In one case there is two metrics, like the bimetric theory, which has pretty much been ruled out by measurements. It's hard to think of a metric theory without it being GR (general relativity), or one of its competitors like Brans-Dicke scalar-tensor theory, which has also been ruled out. See the article in the link above, GR is by far the most suported. And there is no GR equivalent that is denoted or described as a different kind of metric field theory.
So, what does Wilczek mean? He means a different way of thinking of it. In canonical quantization of gravity the approach was to take the metric components, separate space and time like in the GR ADM formalism (one takes a time coordinate as a starting hyper-surface and evolves the metric). This is basically a Hamiltonian treatment of the GR metric as fields. The quantization never worked out, see the wiki article on canonical quantization at https://en.m.wikipedia.org/wiki/Canonical_quantum_gravity
But see a treatment of classical GR as a field theory in http://www.reed.edu/physics/faculty/wheeler/documents/Classical%20Field%20Theory/Class%20Notes/Field%20Theory%20Chapter%204.pdf.
It is not hard to see it as a field theory, and in fact the theory can be set up as a field defined by an action with a Lagrangian, which when the action's variation is set to zero held the GR equations. In fact that is how a scalar field can be tried to be added to the theory, leading to Brans-Dicke. But it all results in the same equations.
Wilczek is too good a physicist, having gotten the Nobel Prize and all his work in QCD, not to have meant something. It seems that he meant that thinking of it and treating it as a field can make obvious us the similarities with the electromagnetic and other fields in physics, and that quantization always needed something different in going from QED to electroweak unification, and the different quantization in QCD, eventually all led to the standard model (SM), all based on quantum fields. But reading his article it seems to be more than that. Just nothing really specific.
He makes a beautiful case, with good physical examples, though with no mathematics, for space as effervescent (bubbling with excitations), substantial (lots of quantum fields filling it), weighty (dark energy), and elastic (trajectories bend).
There is nothing wrong with using geometry to describe the math for GR, and maybe if space (or spacetime) is all he says it is, some geometrical interpretations can make sense for quantum gravity. He is saying the field theory view of it can be equivalent. Keep in mind that AdS/CFT says that a string theory in AdS spacetime is equivalent to a conformal field theory on its boundary.
But other than the beautiful thought of some unification of GR and quantum theory, I don't see any deeper implication, nor any prescription or thought as to how to arrive at it, from what Wilczek said in your reference.
I am open to having missed something deeper or more specific in an answer to the questioner.