The correct answer is threshold.
A threshold is a point or level at which something begins or changes. In this case, it refers to the point at which young people become adults. Many of the advances of civilization have been conceived by young people just on the threshold of adulthood, because they are at a point in their lives where they are full of energy and ideas, but they also have the maturity and experience to start to put those ideas into action.
The other options are not as relevant. A horizon is a line that appears to bound the visible extent of the Earth or sky. A boundary is a line that marks the limit of an area or territory. A peak is the highest point of a mountain or hill.