The following theorem is taken from Matsumura's Commutative Ring Theory [M] Theorem 7.3(i) and the paragraph before it. My questions only concern the proof of the Theorem below.
A ring homomorphism $f:A\longrightarrow B$ induces a map ${}^{a\!}f:\mathrm{Spec}(B)\longrightarrow\mathrm{Spec}(A)$, under which a point $\mathfrak{p}\in\mathrm{Spec}(A)$ has an inverse image \begin{equation*} {}^{a\!}f^{-1}(\mathfrak{p})=\{P\in\mathrm{Spec}(B):P\cap A=\mathfrak{p}\} \end{equation*} which is homeomorphic to $\mathrm{Spec}(B\otimes_{A}\kappa(\mathfrak{p}))$.
Theorem. Let $f:A\longrightarrow B$ be a ring homomorphism and $M$ a $B$-module. If $M$ is faithfully flat over $A$, then ${}^{a\!}f(\mathrm{Supp}(M))=\mathrm{Spec}(A)$.
The proof of the theorem given by [M] is as follows:
For $\mathfrak{p}\in\mathrm{Spec}(A)$, since $\kappa(\mathfrak{p})\neq 0$, we have $M\otimes_{A}\kappa(\mathfrak{p})\neq 0$. Hence, if we set $C=B\otimes_{A}\kappa(\mathfrak{p})$ and $M'=M\otimes_{A}\kappa(\mathfrak{p})=M\otimes_{B}C$, the $C$-module $M'$ is non-zero, so that there is a $P^{\ast}\in\mathrm{Spec}(C)$ such that $M'_{P^{\ast}}\neq 0$. Now set $P=P^{\ast}\cap B$. Then \begin{align*} M_{P^{\ast}}'=M\otimes_{B}C_{P^{\ast}}=M\otimes_{B}\left(B_{P}\otimes_{B_{P}}C_{P^{\ast}}\right)=M_{P}\otimes_{B_{P}}C_{P^{\ast}} \end{align*} so that $M_{P}\neq 0$, that is, $P\in\mathrm{Supp}(M)$. But $P^{\ast}\in\mathrm{Spec}(B\otimes\kappa(\mathfrak{p}))$, so that as we have seen $P\cap A=\mathfrak{p}$. Therefore, $\mathfrak{p}\in{}^{a\!}f(\mathrm{Supp}(M))$.
[M] seems to have skipped a few lines here and there in the proof of the theorem, and I've not been able to see how [M] obtains the following:
Why is $M'$ a non-zero $C$-module? (My guess is that $M\otimes_{A}\kappa(\mathfrak{p})\neq 0$ as an $A$-module, and so is non-zero as a $C$ module. Is this a correct understanding?)
Does such a $P^{\ast}$ exists such that $M_{P^{\ast}}\neq 0$? (I have completely no clue on this)
Any help or advice would be appreciated. I also do not have any knowledge on algebraic geometry, and as such, if there are books that I should look at in this aspect, please do recommend too. Thanks!
For question 1, your understanding is correct.
For question 2, I suppose you mean why is there a $P^*$ such that $M\color{red}'_{\!P^*}\ne\{0\}$. That is because as $M'$ is nonzero, its support in $\operatorname{Spec}C$ is nonempty.