The proposition consists of two parts. Part 1: During the time when the vessels were broken... This explains in general terms what happened to the vessels during the time when they were broken. Part 2: And their source... This explains the difference between the breaking and the repair.
Part 1: During the time when the vessels were broken, the power of government was taken from them. The concept of being broken applies to things that are unfit to carry out their function. This has two meanings that are both really one. For as you have already heard, the Sefirot are the underlying measures of all that exists (i.e. of the separate realms and beings) and the Sefirot themselves are also the laws of government. In other words, it is the separate creations that are subject to the governmental order, while the Sefirot are the laws of the governmental order itself. Thus when one part holds sway over the creations, it results in one type of government, and when a different part holds sway, the result is a different type of government.