If there were only a finite amount of primes then there would be some largest prime $p$ However $p!+1$ is not divisible by any number $1<n\leq p$ since $p!$ is, so $p!+1$ cannot be factored by the primes we already know, but every integer greater than one is divisible by at least one prime, so there must be some prime greater than $p$ by which $p!+1$ is divisible.

Actually Euclid did not use $p!$ for his proof but stated that if there were a finite list $p_1,\ldots,p_n$ of primes, then the number $p_1\cdots p_n+1$ is not divisible by any of these primes and thus either prime and not in the list or divisible by a prime not in the list.