Well, the basic gist is that when the observer becomes entangled with the measurement apparatus, there simply is no collapse—instead, the state is interpreted as referring to one observer making one observation, and another version observing something else.
Schematically, it’s somewhat like this.
Before the measurement, we have:
(observer pre-observation) (measurement apparatus in “ready” - state) (electron spin up + electron spin down)
Then the measurement occurs, and we have:
(observer pre-observation) (measurement apparatus indicates up, electron spin up + measurement apparatus indicates down, electron spin down)
This is an entangled state between the apparatus and the electron. Now, the experimenter looks at the apparatus, and we get:
(observer sees up, measurement apparatus indicates up, electron spin up + observer sees down, measurement apparatus indicates down, electron spin down)
In Copenhagen QM, one then invokes the collapse, so that only one of the terms in the sum survives; many worlds then simply keeps the whole state as it is, and interpretes it as two ‘copies’ of the observer making distinct observations in distinct ‘worlds’.