Euclid built on top of the fact that every number is built by multiplying together primes.

Suppose you give Euclid a list of 109 (for example) primes p(1),…,p(109) and claim that every number can be built by multiplying together these primes.

Euclid constructed a number which can’t be built by multiplying together these numbers: he multiplied all the primes in your list and then, the act of genius, he added one to the number:

p(1)x…xp(109)+1

This number can’t be built by multiplying together primes in your list because the number has remainder 1 on division by any of your primes. Some of your primes are meant to exactly divide Euclid’s number if they are the only primes. Since they don’t you must have missed the primes that built this number. But adding these to the list won’t rescue you because given any finite list of primes Euclid will always be able to build another number which can’t be built from that finite list of primes. Hence the only option is that there are infinitely many primes.