With the aim of fleshing out Jeff's link, insight is the broad concept you are talking about. More specific terms which focus on the apparent abrupt shift include Eureka effect and Aha! effect.
The Kounios and Beeman (2009) article mentioned by Jeff discusses potential neural models of the Aha moment. Kounios and Beeman state that
Although the experience of insight is sudden and can seem disconnected
from the immediately preceding thought, [studies using EEG and fMRI] show that
insight is the culmination of a series of brain states and processes
operating at different time scales. Elucidation of these precursors
suggests interventional opportunities for the facilitation of insight.
More broadly, you ask why there is such a threshold for understanding. One answer is that the phenomena by definition only applies to phenomena that operates that way. Most understanding does not involve an Aha moment. Problems are routine, or they involve searching for an answer by reading materials, or they involve asking someone, or they involve working through a process, and so on. Furthermore, most problems do not have single discrete insights that are going to make a huge difference. Rather, successful performance is the culmination of many smaller tasks. All this is just to say, insight by definition involves a particular dynamic of impasse and then discovery.
Some literature on the Aha moment focuses on problem solvig techniques to both explain the occurrence of the impasse and the insight. For example, problem solvers upon reaching an impasse may start to re-arrange knowledge or engage in various restructuring processes. The literature also discusses incubation processes which in theory may permit more distal knowledge structures to be integrated with the problem.
- Kounios, J., & Beeman, M. (2009). The Aha! Moment The Cognitive Neuroscience of Insight. Current Directions in Psychological Science, 18(4), 210-216. PDF