A state like what owns Newcastle, Man City or PSG is in charge. You bring the tribal fan base of the club over to your side. It generates positive publicity. You can invent sponsorships by telling a company who you are literally affiliated with to just sponsor the club for an arbitrary amount. The fans are happy and quiet because the club is successful and has infinite money, and so will defend the owners which in return is defending the country/state. The state in question wants to distract from oppression and human rights violations.
To be clear that is not what PSG or City owners wanted to do and why they purchased Football clubs in particular QSI. The reason QSI bought PSG is because they wanted to diversify their portfolio and PSG was packaged with considerably more valuable assets in real estate in Iles de France and also a pretty large multinational.
Those deals are way bigger than football fans seem to think. Also none of these nations need sportwashing, they are openly "business buddies" with the wealthiest and most powerful nations.