The flag actually dates back all the way to 1580 and the founder Juan de Garay.

The cross that the eagle holds comes from the Order of Calatrava, a military monastic order, and symbolizes the evangelization of the Americas. https://en.wikipedia.org/wiki/Order_of_Calatrava

The four baby eagles symbolizes other cities in colonial Argentina.

The flag has been criticized for its monarchistic and imperialistic symbolism.

Also interesting is that the eagle looks to the sinister (to the right, seen from the front), contrary to heraldic convention.

Wiki: https://en.wikipedia.org/wiki/Flag_of_Buenos_Aires

Official site: https://buenosaires.gob.ar/laciudad/simbolos-de-la-ciudad/bandera-de-la-ciudad

A blog post about the flag: https://heraldicaargentina.blogspot.com/2019/06/el-antiguo-escudo-de-buenos-aires.html