Planning for the LiMux project started in 2002 on the initiative of an SPD city councillor. It was about migration from Windows NT to Linux rather than a newer version of Windows. SPD (social democratic party) usually has the majority in the Munich city council, and had it at the time. However, in the 2014 elections SPD has lost 8 seats and CSU (christian social union) has won 3. CSU now has 1 more seat than SPD.
Microsoft obviously never liked LiMux and tried to prevent it by offering very cheap licences. The city decided to carry out the project anyway. After some initial difficulties the project was successful and saved the city a lot of money. While it's hard to assess how much money was saved exactly, this should give an idea: Currently half the computers used by the city administration have a CPU frequency of 500 MHz or below, and most have 256 MB of RAM or less. But today CSU is stronger, and Microsoft has gained additional leverage by moving its German seat to Munich (from nearby Unterschleißheim) - a decision in which the current mayor (SPD) was involved.
Part of the reason CSU gets support from computer users in the city administration is that the users do not have administrator rights on their computers. This is of course by design rather than a defect of Linux. However, it is a defect of Windows that large organisations often have to grant administrator rights to their users because often the simplest things don't work properly on Windows without them, with no reasonable workarounds that don't involve a lot of work by system administrators. So in a sense the users are right to complain about Linux: It prevents them from getting rights they shouldn't have on their work computers in the first place!