In the many worlds interpretation, measurement devices (i.e. including things like the "conscious" (whatever that means) you) are equally part of the quantum system along with the putative "quantum system" that is under your measurement.
During the "measurement", nothing remarkable happens: the whole quantum system (i.e. you and the studied system) remain in a pure (albeit monstrously high dimensional) quantum state. The state of this composite system evolves unitarily and utterly deterministically. The "you" and "measured" subsystem become entangled by the measurement, but this entanglement is still an outcome of deterministic pure state evolution.
So the point is that you and the measured system never come out of quantum superposition. All the base states (different "Worlds") are still present and in quantum superposition.
Now, as to which "World" "you" "see", that is quite another matter, and one which science has no accepted answer for at present. This is because science has no accepted precise description for consciousness and thus no description for your (nor mine, nor anyone else's) subjective experience; it has no consistent conception of "you", "I", nor any other personhood.
So the simple answer to your question is that "science does not know". There are workers actively thinking about these ideas, but nothing has yet emerged as being remotely like "scientifically accepted".
User Innisfree comments / asks:
This is nothing to do with consciousness etc. The question is really, how to derive probabilities of observing various outcomes in QM (i.e Born rule) in MWI?
This is an insightful reinterpretation of the OP's original question and one we can make some possible headway on. I am not an expert in MWI, but I understand that something approaching a possible explanation for the seeming emergence of a small number of possible states from a measurement even though the whole system evolves unitarily and deterministically is as follows. The interaction of the studied system with an external system will force the whole to undergo deterministic, unitary evolution determined by a Hamiltonian of the form $H_S + H_M + H_I$, where $H_S$ and $H_M$ are the Hamiltonians of the studied and measurement systems in isolation and $H_I$ is an interaction term. As outlined by user Daniel Sank in his answer here, some models of the interaction Hamiltonian $H_I$ reproduce the interesting behavior that, if the measurement system has a large number of unknown degrees of freedom, then then the result of the unitary evolution tends to be towards a pure state wherein the studied system part is near an eigenvector of a certain Hermitian operator, one of the so called tensor factors the interaction Hamiltonian $H_I$. Different eigenvectors are "selected" (so called einselection) by different initial states of the measurement system. The "probability" of selection of each eigenvector in this model (i.e. the proportion of unknown measurement system states, assuming each is equally likely) is given by the Born rule. But take special heed that there is no randomness here and no wavefunction collapse. This is a unitary state evolution brought about by the presence of the composite system.