Confusion about how the many worlds interpretation denies the measurement?

If you take an electron microscope and measure the position of an electron, quantum mechanics predicts a range of possible positions, governed by its wave function and the uncertainty principle. In the many-worlds interpretation, each such possible position splits off its own universe in which it is the actual position. When you measure it in that particular position, all you are doing is discovering which of those many split worlds you have followed. There are now many other parallel yous who measured different positions.

MWI doesn't deny the existence of measurements. What it does is deny that any special postulates or physics are needed to explain measurement. Systems just evolve according to the Schrodinger equation, whether or not they are being measured; measurement is a socially defined term for a subset of system evolutions contrived by humans to gain information about systems being measured.

Under textbook QM systems evolve according to the Schrodinger equation except when measured, when they collapse to one of their eigenstates with a probability given by the Born rule. This is clearly very unsatisfactory as a vague high level concept like 'measurement' cannot be defined or distinguished in terms of fundamental physics.

For MWI this collapse does not occur; when I do a measurement I do indeed see only one eigenstate with a probability given by the Born rule, but that because 'I' am one 'I' on one branch and there are other 'I's on other branches seeing the other eigenstates. The most interesting open questions (for me anyway) around the philosophical aspects of MWI are around how we make sense of this fundamentally indexical notion of 'worlds', as well as explicating how probabilities work here.