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You are paying for data stored. Which is basically inputs, outputs, and headers.
Please have a look on the schema above.
Output B (of original TX) is the same as input B (of CPFP TX). When you remove them you save on fees. Both are redundant.
And you don't need even the second TX header since it will be one transaction now.
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Fee rate has to be increased a bit to motivate nodes to broadcast the new transaction. But total fee can be still lower. See the schema where I'm removing B output, B input, and one header. Or the example from app print-screen is real case where I save 16% even with increased fee rate.
The bigger fees the bigger saving...