That part is not mysterious either - the collapse simply doesn't occur.
Say you have a photon in a superposition of state A and state B - think position. If it is in state A it hits a detector, if it is in state B it does not. If the detector is hit, it displays 'HIT' on its screen - otherwise it does not. If 'HIT' is displayed, the researcher thinks photon was in state A, otherwise the researcher thinks it is in state B.
The result of the experiment, without any wavefunction collapse, is a superposition of two states of the entire system:
State 1: Photon is in A, Detector was Hit, Researcher thinks photon is in A
+
State 2: Photon is in state B, detector was not hit, researcher thinks photon is in B.
As you can see, the fact that the researcher never sees a photon in a superposition of states a+b (he doesn't ever see the detector both lit up and not) is explained without any wavefunction collapse.
Say you have a photon in a superposition of state A and state B - think position. If it is in state A it hits a detector, if it is in state B it does not. If the detector is hit, it displays 'HIT' on its screen - otherwise it does not. If 'HIT' is displayed, the researcher thinks photon was in state A, otherwise the researcher thinks it is in state B.
The result of the experiment, without any wavefunction collapse, is a superposition of two states of the entire system:
State 1: Photon is in A, Detector was Hit, Researcher thinks photon is in A
+
State 2: Photon is in state B, detector was not hit, researcher thinks photon is in B.
As you can see, the fact that the researcher never sees a photon in a superposition of states a+b (he doesn't ever see the detector both lit up and not) is explained without any wavefunction collapse.