Assuming it's the april 2012 article "Parenting Adolescents and the Problems of Letting Go" in Psychology Today, the meaning here is "(of) the world".
Here's the entire paragraph from the article.
During the third stage of adolescence, late adolescence (ages 15 – 18) there is the letting go of younger restrictions as some older freedoms (driving, dating, and part time employment, for example) are allowed by parents and others, not necessarily parent approved, that are encouraged by peers (substance use, sex, and adventurous risk taking, for example). This creates a very scary letting go for parents who must accept that they cannot keep their teenager free of more worldly dangers that are associated with the worldly experience he or she is wanting. The loss of parental protection that was provided by older restrictions may be partly supplanted by adequately preparing the high school teenager to understand and manage new risks that come with acting more grown up. Parents have a duty to inform.
So according to the article, adolescents will want to experience the world and to partake in the world's freedoms, but that opens them up to the dangers of the world. The examples of "worldly experiences" used in the article are driving, dating, part time employment, but also substance use, sex, and adventurous risk taking. These experiences come with associated dangers.
As the penultimate line reads,
The loss of parental protection that was provided by older restrictions may be partly supplanted by adequately preparing the high school teenager to understand and manage new risks that come with acting more grown up.
So the article assumes children to live a more protected life up to then, so it's the world in contrast with sheltered family life.
This is not how worldly is usually used.
Merriam Webster defines it as
1: of, relating to, or devoted to this world and its pursuits rather than to religion or spiritual affairs
The article however does not mention religion or spiritual affairs, so the contrast used here is different. Whether that was clumsy writing, or a deliberate choice by the author to evoke the regular meaning, remains speculation.