Easy part first: American sports leagues don't have cash transfers like soccer does, basically bc they have salary caps that lokit what they can spend on players. Details vary by sport but there are 3 main ways a player can change teams
1) trade: this can involve cash but usually is trading players for other players and/or draft picks (unlike Europe where players are developed by team run academies, here they're drafted out of HS/college or internationally)
2) free agency: a player's contract ends up and they're free to sign with any team who wants them. This was not always the case but earlier players (see Curt Flood) fought for it. Sone major leagues have restricted and unrestricted forms and various other exceptions. Important sidenote: every major league has a rookie pay scale related to draft position, and the most valuable thing in all of sports is a star player still on a rookie contract (i.e. way under market value), which usually lasts for 4 years.
3) cut/waivers: a player is released by a team for performance and/or behavior reasons and can sign with any team that wants them. In some sports - particularly the NFL - a lot of the $ in a contract isn't guaranteed, so if a player gets cut the team isn't on the hook for it.
Salary cap management, esp as it relates to trades, is a complex thing. It might interest you as a former finance guy.