The question is badly (!terribly!) worded.
It is asking how many elements $P(\emptyset)$ has. And $P(\emptyset)$ has $2^0 =1$ element.
As every element of $P(A)$ is a subset of $A$ then the question thinks it is asking. "Consider $P(A)$; how many subsets of $A$ are there that are elements of $P(A)$". And the answer to that is $2^{|A|}$.
But by saying "subset" is not clear whether it is asking how many subset of $A$ are elements of $P(A)$ (which would be $2^{|A|}$) or whether it is asking how many subset of $P(A)$ are there (which would not be elements of $P(A)$ but elements of $P(P(A))$. If interpreted this way the answer is $2^{|P(A)|} = 2^{2^{|A|}}$.
$\emptyset$ has exactly one subset (of $\emptyset$). It is $\emptyset$.
So $P(\emptyset) = \{\emptyset\}$. And $P(\emptyset)$ has one element. Its one element is a subset of the emptyset. So $P(\emptyset)$ has one subset of the emptyset as an element.
But $P(\emptyset) = \{\emptyset\}$ has two subsets of itself. They are $\emptyset \subset \{\emptyset\}= P(\emptyset)$; and $P(\emptyset) = \{\emptyset\}\subset \{\emptyset\} = P(\emptyset)$.
......
So to my mind I would think the correct answer to "The power set of the empty set has exactly ------- subsets"; I'd assume the question is asking how many subsets of itself exist. And the answer to that is clearly: $2$.
But that is not what the question intended. The question intended "The power set of the empty set has exactly ------ subsets (of the empty set) as elements". And the answer to that is $1$.
In my opinion the fault is entirely on the authors for writing a !terribly! worded question.
......
(Or possibly the authors of the answer key were not the people who wrote the questions. And the authors of the answer key didn't know the answer and they misread the question and gave the wrong answer.)
Anyway....
if a set $A$ has $|A|$ elements then it has $2^{|A|}$ subsets. So $P(A)$ which is, by definition, is the set of all subsets, will have $2^{|A|}$ elements.
So $\emptyset$ has $0$ elements so it has $2^0 =1$ subset.
So $P(\emptyset)$ has $1$ element so it has $2^1 = 2$ subsets.
I think you've got that (thoroughly by this point I hope) so I'll go and bury this dead horse.
Best Answer
As noted in the comments, $P(\emptyset)=\{{\emptyset\}}$.
$P(\{{\emptyset\}})=\{{\emptyset,\{{\emptyset\}}\}}\ne\{{\{{\emptyset\}}\}}$