Here, "the edge" is a metaphor. It is used to evoke an image of pushing oneself to the point just before failure to achieve a goal, in other words, to the point just before falling off of an actual precipice.
Your source material provides an explanation of "the edge". It says:
It [the so-called "t-type"] indicates how comfortable a person is with taking chances—living on the edge. What Farley calls a big T personality is a person who likes risk.
So, living on the edge means living in a way that brings a lot of risk. Your document also lists exemplary activities: "bungee jumping, skydiving, extreme skiing," among others.
As an example, the woman you quoted who likes to go "right up to the edge" could be skateboarder who always tries harder and harder tricks. She would be likely to fail by always pushing right up to the edge of what is safe or what her capabilities allow. On the other hand, when she is able to push right up to the edge and succeed, she would feel a stronger rush than if she accomplished a simple trick.
While metaphors can be hard to figure out from dictionaries, you might try looking into the phrase "on the cutting edge," which is sometimes used when you want to describe significant new advancements in products/technology.