Consider a (nonempty) finite list of primes $p_1, p_2 \ldots p_n$. Since $N := p_1p_2 \cdots p_n +1 > 1$, there must a prime $q$ dividing $N$ and its easy to see that it is not in the given list.
Prime Factorization of integers plays an important role in the above argument and this ultimately rests on Euclid's Lemma: if $a,b$ are integers and $p$ is a prime dividing $ab$ then $p$ divides either $a$ or $b$.
Here is a modified version of classical proof. Choose any positive integer $n$. Note $n$ and $n+1$ are coprime and so $n(n+1)$ has at least two distinct prime factors. Similarly continue the argument with $n(n+1)$ and $n(n+1)+1$ and so on.
