As Abramovich was running out the clock without spending a penny, they probably had some room on FFP figures. They knew they were losing Rudiger and the other CB ages ago, long before this takeover. This - afaik - means the knew owners write off both the book value/amortization and drop the existing wages from the balance sheet. Also consider that they'd have been looking at a balance sheet containing a sizable increase in salary for both. Rudiger will be paid very handsomely indeed by Real.
Money will have been put aside back then for initial payments, and the wages won't be much bigger. Big fees aren't a disaster going forward as they are written off bit by bit. If anything is said, it's likely to be noted that it's a one off season for them.
Short answer - Wages. That's how. In addition to Christensen and Rudiger, they have shipped Werner, Lukaku. Hudson Odoi, Havertz, Emerson. Abraham, Azpi, Malang Sarr. Bakyoko and Alonso, both set to leave today or tomorrow. Rahman strongly linked with a move.
Barclay and Drinkwater both reached end of contract. Niguez end of loan.
City got fees for young players this year - but Haaland's salary is a big expense. Chelsea have signed more new players, but none on that sort of wage. Meanwhile, they shipped a tonne of players who were first teamers, all of whom were getting the big money. Niguez, for example, '10m a year. They've amassed a large ongoing saving, and spent it on fewer and better players to complement the Mounts and James's - both of whom earn less than half what Niguez did, afaik they earn less than any of the others listed above.