What do physicists mean by a topological quantum gravity theory This is a jargon-like question.
The fact that this is posted here rather in a physics forum indicates two things


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*I know too little physics.

*An explanation with more mathematics flavors will be appreciated more..


Background
I should first explain what a gravity theory is in my imagination: it seems to be a theory that governs the relation between the space and the matter. For example, Hilbert dealt with this problem by introducing a functional on the space of metrics, a consequence being Einstein's field equations that relates the curvature of the space-time and the mass-energy-momentum tensor.
A quantum field theory seems (to me) to be a field theory where each field could possibly happen. In our case, the fields are the metrics, whose amplitudes can be computed by some "quantized" action weight
Question
This leads the confusing part: a topological theory (seems to) mean a theory that does not depend on the geometry (in particular, metric)! What does a topological quantum gravity theory mean then?
 A: Physicists here. The input for a physical theory is always some topological space and some structure (such as a metric) that depends on the specific context. The dynamics are invariant under the isometries thereof. For example, the theory of Special Relativity deals with a manifold of the form $\mathbb R^n$, and with a (pseudo)metric $\operatorname{diag}(-1,+1,+1,\dots,+1)$. The dynamics are invariant under the so-called Poincaré transformations, i.e., the group of isometries of the metric above.
We typically think of gravity as a manifestation of a non-trivial geometry, i.e., a generalization of Special Relativity where the manifold and the metric are no longer necessarily of the form above. There are two layers for a theory that includes gravity:


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*Gravity as a background field, where the manifold and the metric are fixed, and the dynamics correspond to other degrees of freedom propagating in this manifold, and

*Gravity as a dynamical field, where the metric (and possibly the topological space itself) is determined by some dynamical equations. The system is to be determined by solving a self-consistent set of equations that include the metric, and the rest of degrees of freedom, each influencing each other.
The former doesn't have a specific name as far as I know; we just call it "dynamics in curved spacetime". The latter is known as a "theory of gravity", the prototypical example being General Relativity and its extensions. Here the metric is determined by a set of PDEs. This system of equations is invariant under diffeomorphisms, as couldn't be otherwise. This is regarded as a generalization of the statement that the dynamics are to be invariant under the isometries of the metric, but now we allow any possible map, not only an isometry (because there is no fixed metric to begin with). This is also known as general covariance.
The epithet "quantum" refers to the fact that the dynamics are, well, quantum. There is no perfectly convincing definition of what it means to be quantum (cf. this physics.SE post), but the general sentiment is that the state of the system is described by a vector in some Hilbert space (as opposed to a classical system, where the state is described by some point in some fibre bundle over your manifold).
A "quantum theory of gravity" is, thus, a model of a system where we include gravity (non-trivial geometry/topology) in a quantum mechanical way. Whatever the model is, it is to be general covariant. A standard way to construct such a model proceeds as follows:


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*First construct a quantum mechanical model that depends on a fixed background metric. We know how to do this, at least in a formal way (that is perfectly good for our purposes).

*Integrate the previous object with respect to all metrics, whatever that may mean.
The latter step guarantees that the result is general covariant. Unfortunately, we don't really know how to do that in practice; any attempt has failed.
Witten (https://projecteuclid.org/euclid.cmp/1104178138) proposed an alternative method to construct a quantum theory of gravity: instead of integrating over all metrics, set up a model that does not depend on a metric at all, from the very beginning. The dynamical variables are typically differential forms, and we only admit operations that do not require a metric (exterior differentiation). Models that are metric-independent are known as topological, because they only depend on the manifold as a topological space (typically with some extra structure, such as a framing or a spin structure, etc.).
So, to sum up: a theory of gravity is a theory where the physical manifold is a dynamical variable itself. One can accomplish this by introducing a metric and allowing it to interact with (and feel the back-reaction from) other degrees of freedom. Another way is to not introduce a metric at all, and use degrees of freedom that can be defined without reference to a metric, such as differential forms. Making the theory "quantum" is still an open problem, and we don't really know what we want here: what does it even mean to have a quantum theory of gravity? what should we ask of such a model? Integrating over metrics is very problematic, while topological gravity is perfectly well-defined, even if very unrealistic from a physical point of view. Perhaps we should use it as a toy model to explore what are the properties of quantum theories of geometry/topology without the noise caused by other more realistic models.
