Proof that perfect privacy implies that the number of keys is at least the number of messages
Let $K$ be the set of all possible keys (for AES-256, this set has $2^{256}$ elements.)
Let $M$ be the set of all possible messages (for AES, it has $2^{128}$ elements).
Let $C$ be the set of all possible cipher texts (for AES, again $2^{128}$ elements).

I was reading a proof to the statement:

Perfect privacy implies that $|K|=|M|$


Actually this is $|K| \ge |M|$.
Perfect privacy condition is: 
that for all $m$ and $m′$, $P_{k \in K}[e(k,m)=c]=P_{k \in K}[e(k,m′).=c]$
the probability that a cipher text $c$ is decrypted as $m$ is the same as being decrypted in $m'$.
Or in other words:
you can't find a difference between $m$ and $m'$.
In order to prove such statement you use reductio ad absurdum. Here you assume $|M| &gt; |K|$.

The first thing that I don't understand is the following notation: $p_{k \in K}[e(k,m)=c]&gt;0$. What exactly does this mean ?

Yes the $p$ is a typo and should be $P$. It means that there exists $k$ and $m$ such as $e(k,m)=c$ (which is trivial because we need a working algorithm...). Therefore the probability that given a $c$, $m$ is the plain text is at least $\frac{1}{|M|}$ (assuming a uniform distribution), hence $&gt; 0$.
About this part:

Therefore $P_{k \in K}[e(k,m′)=c]=0$
Does this mean that the probability that we encrypt $m′$
using $k$ results in the cipher $c$ is equal to $0$ because, as said before in the proof, $m′$ is a message that has no key $k$ that transforms it into $c$? If yes, I am still not 100% convinced.

The first thing you must think of is that the $c$ is fixed. In order to encrypt a message $m$ to this $c$ you need a key $k$. Because $|M|$ is greater than $|K|$ (assume $|M| = 5$ and $|K| = 3$), there exists messages that won't have a mapping to $c$ (in this case $5 - 3 = 2$). Therefore there exists $m'$ such as there is no $k$ so that $e(k,m) = c$. Hence $P_{k \in K}[e(k,m′)=c]=0$
Because the statement of perfect secrecy is for all $m$ and $m'$ you can't have at the same time all probabilities being $&gt; 0$ and $= 0$. Therefore you have a contradiction.