What is this beef between the mayor and the council?

From an Atlanta Journal-Constitution article

Five of the seven members of South Fulton’s city council sued to throw Mayor Khalid Kamau out of office — in part, for allegedly recording closed-door executive sessions for his “personal benefit.” The 98-page lawsuit says Kamau’s conduct prevents the council from effectively governing. During executive session at a November 2022 meeting, council members discovered Kamau was recording it on a cell phone, the suit says. During a Facebook Live broadcast in December, Kamau said he would continue to do so and release those discussions to the public. He has since refused to stop recording executive sessions and further threatened to use the recordings to sue the City Council and or its members, according to the filing.

In addition to seeking Kamau’s removal from office, council members want an injunction or restraining order to keep him from recording executive sessions and making those discussions public. “To be clear, the City Council has only ever used its executive sessions as authorized by the Georgia Open Meetings Act, which makes it unclear how, or why, the Mayor would seek to use these recordings in a lawsuit against the City Council,” the request says.

Georgia law allows public officials to discuss property transactions, potential litigation and employee matters in private. Council members say they can’t effectively do so under the threat of being recorded. “In short, the Council Members are being held hostage by the Mayor; forced to choose between either betraying the City’s confidence and allowing its confidential information to be recorded and disclosed, or simply not addressing these important matters at all,” the suit says.

Did the council have something to do with the trespassing/burglary charges?

No, but the majority of the council members want him removed from office, and the person taking over for now is one of those members. So I assume they're happy. Good ol' political drama