$\bf 1.$ The Axiom of Choice
Given a set $S$, to say that $S$ is not empty is to say that $\exists x(x\in S)$ (in English: there exists some $x$ such that $x$ is an element of $S$). First-order logic has an inference rule which allows us to move from $\exists x(x\in S)$ to some [new] constant symbol $s$ such that $s\in S$.
This process, called existential instantiation, allows us to move from "the set is not empty" to "here is an element of the set". And this is, in effect, means that we [or rather the inference rule] chose some arbitrary element from $S$.
Suppose now that you have $S_0,S_1$ and neither is empty, then you apply the process twice, and you have two elements of $S_0$ and $S_1$ respectively.1 But what happens if we are given some indexed family $S_n$ for $n\in\Bbb N$ and the information that neither of the $S_n$ is empty?
We cannot use existential instantiation infinitely many times. Remember that mathematics, formally, is always on its way to prove something. Proofs are finite in nature, so you can only apply the inference rule for finitely many of these $S_n$'s.
And to solve this we use the notion of a "choice function". If we had a function $f$ such that $f(n)\in S_n$ for all $n\in\Bbb N$, then we wouldn't need to apply any existential instantiation on the $S_n$'s, since $f(n)$ would already be some fixed element of $S_n$. And the axiom of choice asserts that such $f$ exists, in the broadest way possible, namely if we are given any indexed family of non-empty sets (regardless to the index set), then it has a choice function.
Now we can apply existential instantiation to the set of choice functions, which we have proved to be non-empty using the axiom of choice, and obtain the wanted function.
In simple words, if so, the axiom of choice says that given any family of non-empty sets $S_i$ for $i\in I$, there exists a function such that $f(i)\in S_i$ for all $i\in I$.
But of course, this doesn't quite explain the joke in that last panel. For this we need to talk about...
$\bf 2.$ The Banach-Tarski Paradox
The axiom of choice is extremely useful, and it seems extremely natural as well. If we are given non-empty sets, then there is a way to choose an element from each set. But the consequences of the axiom of choice can be counterintuitive at first.
One of them, called the Banach-Tarski paradox, states that given a ball in $\Bbb R^3$, we can partition it into five parts, move these parts around without stretching or skewing them, and then reconstruct two balls each one exactly the same as the original ball.
This is truly mind boggling, and a lot of people object to the axiom of choice on the ground that this process shouldn't be possible. But those people often mistake mathematical balls to actual physical balls (or vice versa) and a non-constructive mathematical process with what we can do by hand [or robot] in real life.
The XKCD that you link is playing exactly on that. The character in the last panel has cut through the pumpkin several times, and suddenly there were two pumpkins. Just like in the Banach-Tarski paradox. And the "narrator" character points out that they shouldn't have used the axiom of choice to carve out a pumpkin.
- This is not a complete account of the events, and there are more issues to care about. But I find that getting into them can be confusing, and initially it is a good idea to think about the problem as repeating instantiation.