How can I understand these two equations about the probability of measurement?

I have trouble understanding these two equations in the nielsen-and-chuang textbook. Suppose we perform a measurement described by the operator $$M_m$$, if the initial state is $$|\psi_i\rangle$$, then the probability of getting result m is:

$$p(m|i)=\langle\psi_i|M_m^\dagger M_m|\psi_i\rangle$$

The form of this equation looks like the overlap between two states, but I'm not exactly sure what does $$M_m|\psi_i\rangle$$ mean? Is this relevant to the projection operator?

Also, given the initial state $$|\psi_i\rangle$$, the state after obtaining the result m is

$$|\psi_i^m\rangle = \frac{M_m|\psi_i\rangle}{\sqrt{\langle\psi_i|M_m^\dagger M_m|\psi_i\rangle}}$$

Why that's the case? Thanks!!

Cross-posted on quantumcomputing.SE

The measurement is a projection so does not preserve the norm. Thus, if $$M_m$$ takes $$\vert \psi_i\rangle$$ to the unnormalized state $$\vert\phi\rangle := M_m\vert \psi_i\rangle$$, its normalized version is \begin{align} \vert\psi_i^m\rangle &= \frac{\vert \phi\rangle}{\sqrt{\langle \phi \vert \phi\rangle}}=\frac{M_m\vert \psi_i\rangle}{\sqrt{\langle \psi_i\vert M_m^\dagger M_m\vert \psi_i\rangle}} \end{align}