The sellers lose from not having all the money up front.
They may also lose money from the hassle of processing multiple transactions (e.g. when customers forget that their credit card expired half way through the rental period, and the payments stop until they are chased up to fix the problem, or else the rental has to be cancelled).
They may also be losing money because of higher fees for more small transactions.
With Sweetwater’s own finance scheme for example, they simply recover those costs from the customer - the monthly payments add up to more than the one-off price, except for special promotions. For some of their finance deals they also pull a credit card reference, which may affect your credit score for other financial transactions.