[BjoernKW]

FIFA only nominally is a private, officially non-profit (yet actually very much for-profit), entity.

They're embroiled in politics at the highest level because football / soccer arguably is the world's most popular (spectator) sport, which is something politicians gladly make use of to promote their own agendas and careers.

FIFA World Cups more often than not involve large public infrastructure programs.

FIFA and its high-ranking officials on the other hand profit in various ways, too, from "mere" tax holidays to downright corruption.