It's often used to describe something that is happening under the auspices of a larger entity, like a government, organization, or person.
Here are some examples:
* "The event was held under the aegis of the local arts council." (The arts council was sponsoring and supporting the event.)
* "The research project was conducted under the aegis of the university." (The university was providing resources and oversight for the project.)
* "The peace talks were held under the aegis of the United Nations." (The United Nations was facilitating and overseeing the talks.)
The word "aegis" itself comes from Greek mythology, where it refers to the shield of Zeus, the king of the gods. This connection gives the phrase "under the aegis of" a sense of protection and authority.