The admins know the exact reason that you got banned and the report/reports filled against you. If they want, they will reply here with more info about what got you banned.

Users can get banned years after they posted something that was copy/pasted. So the fact that you wrote some posts, they remained visible for a long time, doesn't mean that you got banned for one of your most recent posts. In theory, it can be the first one you ever wrote. Admins usually react and assess reports sent by other forum members. If there is no report against you, no actions will be taken my admins most of the time.   

Posts are reported to moderators who check out the report. If the poster needs to be banned, the moderator sends a ban report up to a global mod or admin. The global mod or admin handles all of the ban reports they get at around the same time. Either the admins or global mods don't need to check every account because they trust the moderators to have already done so, or they check quickly because every such report contains references and links to the plagiarism post and to the source text so checking takes little time.