We know that power set of $\{a\}$ is $\{\emptyset,\{a\}\}$. Similarly, I tried reasoning the fact that power set of $\{\emptyset\}$ should be $\{\emptyset,\{\emptyset\}\}$, but this doesn't give me any intuitive sense.
Can someone help me to understand what exactly does it mean?
On
Power set of a null set consists of only one element: Null set. Since null set is subset of every set. But your interpretation is not correct, {∅} is not null.
On
In the title you ask about the powerset of the null set. Since the null set is $\emptyset$, the title thus asks for $P(\emptyset)$.
But in the body of your question you are asking about $P(\{ \emptyset \})$, which is something different. Indeed, $\{ \emptyset \}$ is not the null set, but is the set that contains the null set as its only element.
Maybe you didn't realize that $\emptyset \not = \{ \emptyset \}$?
Be careful, the power set of $\emptyset$ is not the same as the power set of the set that only contains $\emptyset$.
The power set of $\emptyset$ is the set that contains all the subsets of the empty set, and the only possible subset of the empty set is itself, so $\mathcal{P}(\emptyset)=\{\emptyset\}$. But, the power set of $\{\emptyset\}$ is the set that contains all the subsets of $\{\emptyset\}$, and those are both the empty set and the set that contains the empty set. Therefore, $\mathcal{P}(\{\emptyset\})=\{\emptyset\,, \{\emptyset\}\}$.
I hope I didn't confuse you too much.