Tom's answer is good. I am adding a second one, for the reason that some textbooks do not require covering maps to be surjective. (Added one day later: indeed, Hatcher does not require this.)
The first observation that came to my mind is that if these spaces are not path connected it is bad to notationally suppress the choice of basepoint (e.g. to write $\pi_1(A)$ and not $\pi_1(A, a_0)$ for some fixed and explicitly chosen $a_0 \in A$). This quibble would apply equally well to $X$ itself. So let us bring basepoints to the forefront to make things clearer.
I think $X$ is implicitly being assumed path-connected and locally path-connected (or at least assumed something--- it is not just a random topological space), for the reason that some hypothesis is necessary in order to know that $X$ has a simply-connected covering space in the first place. I don't have Hatcher nearby but I think the construction he gives for $\widetilde{X}$ does have hypotheses like this on $X$.
So far this is just a notational quibble. If you assume basepoints have been chosen in the background so that everything is well-defined (as is often done in situations like this) there is still the issue of whether the hypothesis on $A$ is necessary. With the caveat that this is just based on a casual reading, not hours of thought, my feeling is that probably the hypothesis on $A$ is not necessary, but that there is nothing gained by considering more general $A$.
Let me make my suspicion more explicit.
Suppose $A$ is not path connected. You've chosen some basepoint $a_0$ in $A$; let A' denote the path component of $A$ containing $a_0$. By hypothesis, there is stuff in $A$ that is not in A', but none of this stuff is going to affect "$\pi_1(A)$" (which is of course $\pi_1(A,a_0) = \pi_1(A',a_0)$) at all.
Similarly, when you consider the path-component $\widetilde{A}$ of $p^{-1}(A)$, I think you are implicitly required to choose a path-component of $\widetilde{A}$ that contains an element of $p^{-1} \{a_0\}$. (This goes back to the basic definition of the correspondence between covering spaces of a space $Y$ and subgroups of the fundamental group of a space $Y$. If $f: C \to Y$ is any function at all (in particular if it is a covering map), one must choose basepoints $y_0 \in Y$ and $c_0 \in C$ with $f(c_0) = y_0$ before one can even talk about a homomorphism $\pi_1(C,c_0) \to \pi_1(Y, y_0)$ induced by $f$. So to even talk about the subgroup of $\pi_1(A, a_0)$ of $A$ corresponding to $p: \widetilde{A} \to A$ we must have chosen a point $\tilde{a_0}$ in $\widetilde{A}$ that gets mapped to $a_0$.) With that in mind, since $\widetilde{A}$ is path connected and contains something in $p^{-1}\{a_0\}$, we see that $\widetilde{A}$ actually has to be a subset of p^{-1}(A'). It can't contain things that project down to stuff in A \setminus A' (or we could use path connectedness to draw a path in $\widetilde{A}$ from such a thing to $\tilde{a_0}$, and compose this path with $p$ to get a path from $a_0$ to something that isn't in $A'$).
So all of the data in the statement ends up depending only on the component A' in which the basepoint is chosen, and not on the rest of $A$. I think that is why you never need to use the hypothesis explicitly: all of the loops you discuss when you sketch that argument (suppressing the choice of basepoint) are only taking place in one path component of $A$ (whatever part of it your secretly chosen basepoint lies in). It only becomes mysterious when you suppress the choice of basepoint. Anyway, that is my guess about what is going on.
I guess I have only explained why "path connected" is part of the hypotheses--- not the "locally path-connected" piece. Frankly I can never remember why things like that are added in except that you see it all over the place in discussions of fundamental groups, probably because it rules out pathologies. (Revised on edit: see the comment to Tom's answer. I think Hatcher is using it just because it simplifies some proofs. Probably it is not strictly necessary, but if he were to drop it in the exercises, he would have to give more complicated proofs of the results he does prove in the text.)