Okay kiddo, so a long time ago, France was in charge of a place called Indochina. You know how your teacher sometimes needs help in the classroom? Well, France was like the "teacher" of Indochina.
They had people called governors and high ranking officials called "Residents" who made sure that everything ran smoothly in Indochina.
But the people who lived in Indochina didn't always like being told what to do by the French "teacher." So, the French tried to make it seem like they were helping the people by building things like hospitals and schools.
But sometimes things got pretty tense between the French and the people in Indochina. Eventually, the people in Indochina started fighting back and trying to become independent from France.
After a long time, France gave up and left Indochina. The different parts of Indochina became their own countries like Vietnam, Laos, and Cambodia.
So, in short, the political administration of French Indochina was when France was in charge of a big piece of land and tried to keep things under control, but eventually people in Indochina wanted their own independence.