Exercise frontier represents the decision boundary when you would exercise an option.
This is what a exercise frontier would look like in American option. A more common name is "exercise region". This is the region where it's optimal your option.
L is the optimal exercise price. It's a convex function of maturity. Far away from maturity, the optimal price is significantly lower than K because we'd expect a deep in-the-money intrinsic value to compensate giving up the option rights early. The price approaches to the strike price as shown in the plot because you would have to be less selective for exercising (i.e: you don't have much time to wait).
The concept of exercise region is closely related to optimal stopping time, I recommend Sheve's Stochastic Calculus book if you're interested to learn more.