It's generally financially strapped teams in economically tough markets with unstable ownership who do this. It also for the most happens with players who are close to becoming free agents or are eligible for arbitration and the team knows they can't afford to pay them what they're worth in either scenario. So teams will sell them to get as much as they can instead of losing them for nothing.
And it does happen in basketball, but you're right about it not happening as much in the NFL. The NFL with players having such short career spans and the much higher risk of career ending injuries design their league that greatly favors owners and the ability to keep players much easier for ownership. The NFL even has the "franchise tag" that pretty much gives ownership the ultimate trump card. The NBA doesn't have anything close to that. The closest thing is having the right to match an offer with a restricted free agent. The MLB is more like the NBA in that regard in that it's a player's league and contracts are structured in favor of the players, almost all of them are fully guaranteed contracts. Arbitration somewhat helps ownership to keep the good players but it also favors players from being low balled without recourse.
Teams sell their best players because it's hard to keep them around if the team is losing, because that means the team is losing money and it will be hard to pay the best players what they're worth. Florida, Pittsburgh, Cincinnati are teams that annually sell their best players because they can't afford them. Sometimes the best course of action is to get as much as you can for those players before they walk away and leave the team for nothing but maybe a first round compensation pick.

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