The MWI is what you get if you take the equations of motion of quantum theory and work out its implications as one would for any other physical theory.
You ask:
Or how would you derive the principle of measurement causing a wave collapse?
You wouldn't since collapse is incompatible with the equations of motion of quantum theory.
Instead you would notice that classical physics describes the evolution of a measurable quantity such as the $x$ position of a particle by a function $x(t)$ whose value at time $t$ is the result you would get if you measured the $x$ position.
In quantum physics a measurable quantity is described by a Hermitian matrix called an observable. The possible results of measuring that quantity are the eigenvalues of the observable and quantum theory predicts the expectation value of the observable. In general that expectation value will depend on what happens to all of the possible values during the experiment: this is called quantum interference, see Section 2 of this paper for an example:
https://arxiv.org/abs/math/9911150
When information is copied out of a quantum system interference is suppressed - this is called decoherence:
https://arxiv.org/abs/1911.06282
Decoherence doesn't eliminate the different possible values (i.e. - collapse) it makes them unable to interact with one another so they evolve autonomously. The end result of this is that reality as described by quantum theory looks approximately like a collection of parallel universes in the circumstances of everyday life:
https://arxiv.org/abs/1111.2189
https://arxiv.org/abs/quant-ph/0104033
Decoherence doesn't totally eliminate interference so how you slice up reality into universes isn't exact. In practice two versions of you sitting one millimetre apart won't undergo interference so they are independent, although the electrons in atoms in your body are still undergoing interference on scales of about $10^{-10}m$.
There are a couple of different approaches to understanding probability in quantum theory without collapse. One is to say that the probability of an outcome is a function of the state that satisfies particular properties required by decision theory, like if a person uses those probabilities you can't make him take a series of bets that will make him lower the expectation value of his winnings, the probability of an outcome can't be changed by a later measurement and some other assumptions
https://arxiv.org/abs/0906.2718
https://arxiv.org/abs/quant-ph/0303050
https://arxiv.org/abs/quant-ph/9906015
https://arxiv.org/abs/1508.02048
Another approach called envariance looks at invariant features of the joint state of a quantum system entangled with the environment and using that to work out probabilities using the ignorance interpretation of probability:
https://arxiv.org/abs/quant-ph/0405161
I should also note that quantum theory with collapse isn't equivalent to quantum theory without collapse since the equations of motion of quantum theory don't include collapse, claims to the contrary notwithstanding. Some physicists have tried to change the equations of motion of quantum theory to include collapse:
https://arxiv.org/abs/2310.14969
These theories don't currently reproduce the predictions of relativistic quantum field theories which are the vast bulk of experimentally tested predictions of quantum theory:
https://arxiv.org/abs/2205.00568
Collapse also doesn't describe most measurements, e.g. - it doesn't describe repeated, continuous or unsharp measurements:
https://arxiv.org/abs/1604.05973
