How does the Many-Worlds interpretation really work conceptually? Very often I hear something like: When a decision is made, there is this splitting of the Universes. Each outcome happens but each in a different Universe. So let's take the old Schrodinger's cat experiment. Suppose I decide to make this kind of experiment. Is there an exact version of me in another Universe that is preparing to perform exactly the same experiment, in exactly the same location, with the same conditions etc? Is looking inside the box what causes the splitting? Or this Universe does not exist before my measurement? But if it does not exist, how is it created? Is every measurement creating a new Universe?
Quantum mechanics has a very definite formalism for computing probabilities of interactions. This means one can enter the boundary conditions to the solutions of quantum mechanical equations and get a predicted probability distribution. A specific measurement is one point in building up the probability distribution. In this answer of mine the probability distribution for the double slit experiment single electron at a time is displayed.
There are more than one mathematical formalism for calculating quantum mechanical probability distributions. "Interpretation" comes in in how one reads the mathematical formulae involved: the same mathematics, so it agrees with the usual solutions based on the postulates of quantum mechanics, but interpreted/read differently.
The many worlds interpretation, uses the path integral mathematics of quantum mechanical theory,
The path integral formulation of quantum mechanics is a description of quantum theory that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude.
The usual interpretation is to accept the path integral method as a very useful tool for organizing the calculations of probability distributions for specific interactions in a larger complete format.
The many worlds interpretations, takes this infinity of quantum mechanically possible trajectories as really existing, in an infinitely many worlds universe which exists for always. All possible paths have been already taken.
It is an interpretation that cannot be experimentally checked, because it interprets the same mathematics. It does not predict any differences in the outcomes to be checked by experiment.
How probable it is that it is a good interpretation is left to the mind of the reader, as there are nested infinities involved in every large or small quantum mechanical interaction. Keeping in mind that the classical emerges from the underlying quantum states, the mind boggles.
Unless we allow for magic (that is, events that are not caused by prior conditions), the act of measurement is itself an event that is caused by the events that precede it. The very act of deciding to measure or not to measure is (presumably) a quantum mechanical event which has a probability distribution associated with it: you might decide to measure, or you might decide not to. So the "splitting" of worlds caused by a measurement event is simply a consequence of the state of the world before the measurement event.
Is there an exact version of me in another Universe that is preparing to perform exactly the same experiment, in exactly the same location, with the same conditions etc?
Is looking inside the box what causes the splitting?
Not exactly. The splitting is caused by everything that led up to the act of looking inside the box, including all the quantum mechanical events that led to making the decision to look inside.
Or this Universe does not exist before my measurement? But if it does not exist, how is it created? Is every measurement creating a new Universe?
In the Many Worlds view, if we attribute the decision to look inside the box to natural quantum mechanical processes acting in and on the observer's brain (rather than to a magical nondeterministic "free will"), then the past, present, and future -- along with all their branching possibilities -- coexist. New worlds are not "created" by measurement or observation; they "already" exist as projections of a universal quantum wave function.