The question of how "rapid" learning could be possible relates to Hume's problem of induction -- how can we learn so much from so little. Historically, in both philosophy and psychology, the solution has fallen into one of two camps: either some form of the knowledge was already there to begin with (a 'nativist' view), or we use statistical inference to slowly update our beliefs (some early connectionist models fit into this camp, though in the last 20 years, there are forms of connectionist models that might suddenly/dramatically 'shift' weights in ways that can lead to the appearance of rapid learning).
A solution to the 'rapid learning' problem comes from Bayesianism - the idea that we're doing 'backwards inference' to get to a model of the world. This inference to the best model is weighed by both the likelihood of the data given a model, as well as (and here's the important part) our prior beliefs about how likely the model was before we observed new data. Bayesianism is not just a middle ground between nativism and empiricism; the reason it provides solutions to 'rapid inference' is because you're integrating the idea of 'probabilities' into the mix. That is, rather than there being one absolute truth, we can think of the learner as weighing multiple possible truths - with some a priori belief in each; new data comes in and allows us to rapidly update those possibilities, based on the likelihood of the new data under each.
The Bayesian framework also allows for hierarchical inferences (each set of prior beliefs is given by a more abstract set of prior beliefs) which can tie in nicely to ideas like 'core knowledge' (which fall under the nativism camp), while still providing a learning mechanism (which fall under the empiricist camp).
There is long standing research in vision and language which cover these Bayesian ideas, but in the last decade an explosion of research has come out of labs such as the Tenenbaum lab at MIT, the Griffiths lab at Berkeley, etc. that show how other cognitive/higher-order reasoning (such as causal inference) might be rapidly acquired under this framework. There are several overview papers on these themes which give background on the Bayesian approach, as well as empirical predictions that fall out of specific Bayesian models (many can be found by visiting the lab websites of the aforementioned individuals).
Bayesianism is complementary to models that include additional constraints, such as pedagogical inferences (see Pat Shafto's work) and other pragmatic inferences (e.g. see Mike Frank or Noah Goodman's work) -- all of which to lead to even more rapid inferences from limited data.