This is how I like to think about proper transform. This is somewhat loosely speaking be warned.
Say you are blowing up $Z\subset X$, to get a space $\pi:\hat X\to X$. For a point, say $p$, in the base space, if $p\notin Z$ there is a unique preimage $\hat p$ in $\hat X$ which 'lives above' $p$. On the other hand, if $p\in Z$, then living above $p$ a big chunk of the exceptional divisor, namely the fiber over $p$.
Now suppose you are blowing up $p\in \mathbb C^2$, then the preimage of $p$ is an entire $\mathbb P^1$. On the other hand, say we blowup a line $\mathbb P^1 \cong Z\subset\mathbb P^3$, the entire exceptional divisor is $E\cong\mathbb P^1\times \mathbb P^1$, and the fiber over any point $p$ sees one fiber in $E$.
Great, now in the latter situation, what if you had a curve $C$ passing through $Z$, say intersecting it at a single point. If you just take the set theoretic inverse image you get a curve with a $\mathbb P^1$ attached to it in $\hat X$. This is called the total transform and isnt really what we want a lot of the time.
For this reason, you can follow the following mantra to get the 'right' curve $\hat C$ living above $C$ in $\hat X$. Take $\pi^{-1}(C)$, and 'throw out the exceptional stuff', i.e. set theoretically remove the intersection with the exceptional divisor. Now take closure. This is the proper transform in this case, and you get more what you expect. Instead of this exceptional fiber attached to $\hat C$.
Working out a few examples, like the one above can be quite helpful. You can do this in coordinates even! Take your favourite line inside $\mathbb P^3$ and blow it up, check what the inverse image is, try to figure out what the proper transform looks like.
Best Answer
Take a singularity $p \in C \subset S$, and let $f: S' \to S$ be the blow-up of $S$ in $p$. Let $\tilde C \subset S'$ be the strict transform of $C$. Pushing down the structure sheaf of $\tilde C$ you obtain a short exact sequence of sheaves on $S$ $$0 \to \mathcal O_C \to f_* \mathcal O_{\tilde C} \to F \to 0,$$ where $F$ is a skyscraper sheaf supported on $p$. Using $p_a(C) = 1 - \chi(\mathcal O_C)$, and the fact that the Euler characteristic is additive under short exact sequences, this leads to $$p_a(\tilde C) = 1 - \chi(\mathcal O_{\tilde C}) \stackrel{(*)}{=} 1 - \chi(f_* \mathcal O_{\tilde C}) = 1 - \chi(\mathcal O_{C}) - \chi(F) = p_a(C) - \dim H^0(F).$$ Equality $(*)$ is true because the restricted map $f: \tilde C \to C$ is finite, hence affine, which means that $H^i(\tilde C, G) = H^i(C, f_*G)$ for all quasi-coherent sheaves $G$ on $\tilde C$.
Finally it remains to show $\dim H^0(F) > 0$. As $F$ is supported on $p$, this is equivalent to $F \neq 0$ or that $\tilde C \to C$ is not an isomorphism. I guess one should be able to show that using local coordinates?