The phrase often implies someone is being overly assertive or aggressive in exercising their influence. He throws his weight around in meetings, often talking over others.
Pro Tip 2/3
Negative Connotation
It usually has a negative connotation, suggesting abuse of power or authority. She got the promotion, but now she's throwing her weight around the office.
Pro Tip 3/3
Hierarchy Implied
Use it when referring to situations where there is a clear hierarchy or power dynamic. The senior manager threw his weight around to change the project direction.