Measurement postulate
The statement you are asking about is a postulate of quantum mechanics, so it cannot be mathematically derived from other facts in the theory. Instead, it is justified by its agreement with observation in the sense that it allows us to mathematically describe every measurement we can perform in practice.
Intuitive interpretation
The meaning of $M_m|\psi_i\rangle$ is that this is the unnormalized post-measurement state. The measurement postulate provides a prescription for computing two quantities: measurement outcome probability $p(m|i)$ and post-measurement state $|\psi_i^m\rangle$. Both are derived from $M_m|\psi_i\rangle$.
Specifically, the measurement outcome probability $p(m|i)$ is the squared norm of $M_m|\psi_i\rangle$
$$
p(m|i) = \| M_m|\psi_i\rangle \|^2 = \langle \psi_i | M_m^\dagger M_m|\psi_i\rangle.
$$
The reason it looks like state overlap is because the norm is the defined in terms of the inner product as $\| |\psi\rangle \| = \sqrt{\langle\psi|\psi\rangle}$.
The post-measurement (or collapsed) state is the normalized $M_m|\psi_i\rangle$
$$
|\psi_i^m\rangle = \frac{M_m|\psi_i\rangle}{\| M_m|\psi_i\rangle \|} = \frac{M_m|\psi_i\rangle}{\sqrt{\langle \psi_i | Mm^\dagger M_m|\psi_i\rangle}}.
$$
Relationship to projective measurement
This form of measurement postulate defines a more general type of measurement than the projective measurement. The latter is a special case in which $M_m$ are projectors. In this case, $M_m^\dagger = M_m$ and $M_m^\dagger M_m = M_m$ and we recover the familiar projective measurement. In general though the measurement operators $M_m$ do not need to be projectors.
A prominent example of non-projective measurements that can be described using the general form of the postulate are the POVM measurements. However, even though the general form subsumes the projective measurement, it is possible to implement the general measurement using projective measurement, unitary evolution and auxillary subsystems, see p.94 in Nielsen & Chuang.
