Why have minority groups gone to the Supreme Court to outlaw the at-large system of city elections?
Minority groups have tried to get at-large city elections banned because those elections make it harder for the minorities to get any representation in a city.
In many cities, there are large minority populations, but the populations are not big enough to constitute a majority of the residents of the city as a whole. However, the populations are usually fairly concentrated in an area or areas of the city. What this means is that a district-based election system would make it likely that non-white city council members would be elected. The concentrations of minority voters would end up creating some “majority-minority” districts, and those districts would elect non-whites (or at least would elect people who represented their interests).
In contrast, an at-large system of elections makes it much less likely that minority voters will have any clout. Every election will be conducted over the whole city and the white vote will overwhelm the minority vote. (Of course, this assumes that people will vote based on their race.)
In this way, an at-large system is less likely to allow minorities to elect people who represent them. That is why minorities have challenged this system.