The basics of quantum mechanics are:
1. Any system can be described in a linear "state space", whose basis vectors (roughly speaking) are each possible arrangement of the things being described (every thing's location). However, all possible states include (complex) linear combinations of these. This is usually described as the "wave function", you can plug in any value for coördinates of all your things and get a number back out.
2. States in a closed system evolve according to the (time-dependent) Schrödinger equation, -i h-bar d/dt (state) = E(state). This is sometimes phrased as "energy is the generator of time translation" (note that this phrasing holds true in the Hamiltonion formulation of classical mechanics too, but with different, related meanings). Actually calculating this can be difficult, and you might need to do various tricks such as decomposing in a convenient basis.
No all-encompasing energy field binding and penetrating us needed, though the equation is non-local in the sense that it could depend on the entire world. (In fact, it's quasi-local in practice; energy is usually just a function of where things are (local), and how different that coefficient is on infinitesimally adjacent states (i.e. spacial derivatives of the wavefunction). But this does let changes propagate at any speed, including faster than light. Basic QM is not Einsteinian relativistic. It is perfectly Galilean relativistic in a very nice way though.)
3. If you measure a system, you're going to get results that are probabilistic with the probabilities proportional to the (absolute value of) the square of this wavefunction. After that, the state is reset to whay you measured, and evolution will proceed according to 2.
It comes in when you treat the measurement apparatus as separate from the thing being measured. When you do that, collapse flows naturally from that assumption. It's a valid approach, and very useful, but many people find it philosophically awkward.
An approach that's philosophically easier, but less pragmatic, is to treat the masurement apparatus as part of the system. Now you've got one single enormous quantum system. The math of that is far too complex to actually run, and more importantly, too enormous to hold itself in an unstable equilibrium. It must fall into a more stable equilibrium.
The tricky part is when there's more than one stable state. The measurement apparatus exists in both of those states, and returns opposite results. The states are in local equilibria, and don't exchange significant amounts of state with each other.
You can call that "splitting universes", if you want, though I think that's needlessly messianic. Like the first one, it's perfectly valid. And pragmatically it acts just like collapse. There's a philosophical question about whether the other states "actually exist but can't be reached" or "don't exist at all since we can't prove it except by inference from the fact that this one exists and follows the Schroedinger Equation so closely". That's "philosophical" in the sense that there is no physical difference between the two, but just one of your mental model.
That's a lot of words; sorry about that. But the point is that you don't need a collapse axiom, and if you choose to add one, it's not strange.
True enough, but what else could you expect the experience of being in the Everettian universe to look like? I think it's fair to say that experimental results are at least compatible with reality following the Schroedinger equation without needing a collapse postulate.
Another measure of naturalness is "simplicity", and the Schrödinger equation is a fairly simple expression.
You might argue that it's results don't feel very intuitive, and therefore it is not natural. But then I why you think its intuitive that levers work.