Why does Euclid write "Prime numbers are more than any assigned multitude of prime numbers." In Euclid's Elements Book XI proposition 20 (http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX20.html), Euclid proves that:

Prime numbers are more than any assigned multitude of prime numbers.

I know that this is supposed to say something similar as there are infinitely many primes, but I don't really see this from this wording.

In my mind, this sentence means something like: 

There are more prime numbers than any amount of prime numbers.
You can see Aristotle and Mathematics and Actual infinity.

According to the Aristotelian philosophy, we cannot legitimately "handle" actual infinity; i.e. we have no experience of an infinite "collection" but only of an unlimited iterative process (the potential infinity).

Euclid's statement must be understood in this context : we never have a "complete" infinite set of prime numbers, but we have a procedure that, for a finite collection of prime numbers whatever, can "produce" a new prime which is not in the collection.
I disagree with your interpretation at the end, of what he meant to say. I think if you interpret his statement, you should come up with something similar to this idea. "Prime numbers" changes to "there exists prime numbers", "are more" should be read as "greater than", and finally "than any assigned multitude of prime numbers" meaning then any prime number that actually takes a non variable value. In full my interpretation is this.

"There exists prime numbers greater than any prime number that actually takes on a non variable form."

This is what I 'beleive' to myself he said, further I wonder if part of what he actually said has been removed? Reguardless I think it shows knowledge of the idea of prime numbers as a variable for representing all of them. I think it also is clear evidence of the understanding that there are infinite number of prime numbers.