For reference, here is a quantum logic circuit that performs entanglement swapping:
Alice is the top wire, Bob is the bottom wire. They each create an EPR pair $|00\rangle + |11\rangle$ via H+CNOT operations and give them to Victor (the center two wires). Victor then does a Bell basis measurement (by performing operations that un-make an EPR pair then measuring each wire).
The really important thing to realize is that, after that Bell basis measurement, Alice and Bob could be in one of many possible entangled states. Their qubits might be entangled to agree along the X axis, or to disagree. They also might agree or disagree along the Z axis. Concretely speaking, they could be in the state $|00\rangle + |11\rangle$, or in the state $|01\rangle + |10\rangle$, or in the state $|00\rangle - |11\rangle$, or in the state $|01\rangle - |10\rangle$.
Those four states are all entangled states, but they disagree about how measurements should correlate. Pick any measurement where one of those states predicts the measurement should return YES more often, and you'll find that one of the other states predicts that the measurement should return NO more often. They overlap in a way that leaves you with no bias in expected results; there's nothing but noise to be found.
(Speaking technically, what I'm saying is that Alice and Bob's qubits on their own are represented by the maximally mixed density matrix.)
But then there's still Victor's measurement results. They reveal which entangled case Alice and Bob ended up in. Victor knows their state's X-parity and its Z-parity. All Victor has to do is tell Alice and Bob "Hey, the Z-parity came out 'disagree' so one of you flip your qubit 180 degrees around the X axis to fix that". Do the same for the X-parity, and the problem is solved: Alice and Bob will definitely end up in the state $|00\rangle + |11\rangle$ after the corrections. That's what the last two controlled operations are doing.
Of course, you don't have to fix the problem during the experiment. That can be difficult to do. Instead, you can just skip the corrections but then after the fact use Victor's measurement results to split all the runs into the four possible cases. Then within each of those cases you can ask "Do Alice and Bob's measurement results match the expected kind of entanglement?". Without Victor's measurement results, you can't do the splitting into four cases and all the cases cancel out each others' biases so all you can find is noise.
