All international students are eligible, but it’s a selective scholarship aimed at a particular type of student. A lot of international students (not saying this is OP) come to the US with the intent of finding a job here and staying. This is not for them.
“ The Dartmouth Admissions Office selects students for this special honor who embody the vision of the program, including a commitment to alleviating poverty, a record of academic excellence, and a passion for global issues.
…. Upon graduation, King Scholars are expected to return to their home countries to work on poverty issues”