How do you go from 7 billion Toffoli gates (which consist of 42 billion "two-qubit entangling gates", per the blog post) to merely 1400 logical qubits?

Same way you go from 2 billion CPU operations per second to only ~4000 register bits (not counting ZMM SIMD registers) on a conventional CPU.

The operations all consist of saying, connect these 3 bits and do a reversible operation on them all together. Same as assembly, "add these two registers and store the sum in the first one..." You didn't need to introduce any new bits.

You only need to introduce new bits for steps that cannot be reversibly done, in assembly you get around this by being able to overwrite a register: in quantum, that requires an explicit measurement in the computational basis to figure out how you want to do stuff to zero it; zeroing a bit is not a unitary operation. But if you can encode the circuit in Toffoli gates which are perfectly reversible, you don't have to delete any bits after that encoding (but you may have to introduce extra bits to get to that encoding, like using Toffoli to build “x AND y” requires an extra z bit that effectively gets discarded at the end of the computation when everything is done and nobody cares what that bit holds, but it holds the information you would need to reverse that logical AND).

But yeah it's just number of operations that you need to run the algorithm, versus the number of registers that you need to run the algorithm, they're just two different numbers.

A gate isn't a qubit, it's an operation applied to qubits. You can do more than one operation per qubit.