Well I can answer that. In fact there are various notions of topological strings and background independence. Mostly one means with this a certain modification, or "twist", of the ordinary superstring that projects the string down to its "topological" subsector.
This subsector has only a finite number of states, essentially given by zero modes of the fields, and the correlation functions are "topological" in the sense that they are given by mathematically exact expressions. They depend on parameters (essentially) only holomorphically and are largely protected from perturbative quantum corrections. This allows to expose non-perturbative corrections like instanton corrections and compute them exactly.
Because of this solvabilty and tractability, topological strings are an ideal "theoretical laboratory" for testing ideas, develop computational methods and make explicit contact to nice clean mathematics (eg due to mirror symmetry). So they are mainly a toy model for full-fledged strings, where many properties can be studied in a simpler and often exact way. For example, one can get a handle on "space-time foam" near the Planck scale and see very explicitly the kind of generalized geometries that contribute to the path integral. Or one can determine explicitly the microscopic states of quantum black holes. All this is very concrete and mathematically well-defined, so one can go much further as with ordinary methods.
But there is a price to pay: as said, topological strings are tied to a "simple" subsector of the full-fledged superstring, and they can't say much about quantities which are not in this subsector. So they provide only a partial solution, but the hope is that one can get a lot of relevant conceptional insights by studying just this subsector. Fortunately it is also the physically most interesting subsector, dealing with the low-energy or massless states; the correlation functions one can exactly compute are, for example, the Yukawa couplings or gauge coupling threshold corrections, which are of direct phenomenologial interest. Also the solution N=2 gauge theories of Seiberg and Witten can be direcly related to topological strings, and a lot of insights in the non-perturbative sector of general supersymmetric gauge theories had been gained from this perspective (this is different from AdS/CFT). On the other hand, not surprisingly, topological strings do not say much about the gravitational sector.
One kind of conceptual insights, for example, concerns background independence. This has a specific meaning in this context, and this should not be confused with other notions of background independence. As said, topological strings deal only with a sbsector of the theory, and this means that correlation functions depend only on a subset of all parameters (=vacuum expectation values that determine the compactification geometry). Thus they are independent of the "other" parameters in the theory (including anti-holomorphic parameters), and since these are also part of the background geometry, the theory is thus "partially" background independent. Actually the story is much more complicated in that there IS in fact a dependence on the "wrong" kind of parameters, due to anomalies, but it is a mild one that is very well definded and governed by certain differential equations. It is these equations what is interesting in this context. See
http://arxiv.org/abs/hep-th/9306122 for a concrete discussion.
So far I was mainly discussing the space-time aspects of topological strings, but there are also wold-sheet aspects, and indeed the correlation functions I was talking before are metric-independent also from the world-sheet, 2d point of view. From this perspective there are many relations to integrable systems, matrix models etc.
All-in-all, to repeat: the main message is that the (usually considered) topologial string is a simplified toy model and testing ground for full-fledged strings, for which one can go pretty far, both conceptionally and computationally. As for "new, [experimentally] verifiable predictions", this is not the right thing to ask for in this context.
