Person A: The number of even and odd natural numbers is equal, because they alternate each time. Person B: There are far more even natural numbers. If you double an odd number, it's even. If you double an even number, it's still even.
How do I best explain to a person with only base mathematical knowledge the truth of the matter?
On
The second argument only shows that in the "double naturals", there are far more even numbers than odd ones. The argument says nothing about the naturals themselves.
Modern theory says that there are as many even as odd naturals, because you can associate every even number to an odd one and conversely (add or subtract $1$).
But it also says that there are as many naturals as there are even ones (multiply and divide by $2$) ! And there are as many primes as there are naturals.
"There are more" is not well-defined for infinite sets. I think the real answer is that the question isn't meaningful. (The existence of an injective map is well-defined, but you can debate whether that means 'there are at most as many'.)
You may dispute the latter argument by pointing that "If you take a natural number and add its successor to it, the result is always odd". There are infinitely many functions $f : \mathbb N \times \mathbb N \rightarrow \mathbb N$ that always produce odd or always produce even numbers.