Imagine you are the Count von Count\. You care only about counting things\. You don't care what they are, you just care how many there are\. You decide that you want to collect items into plastic crates, and you consider two crates equal if both contain the same number of items\. Without counting them explicitly, how can you tell if two crates contain the same number of items?
Why not just count explicitly?
I think the answer is, "because we want to teach what a bijection is," but readers might be confused why we're doing this. Maybe some of the flavor text about the Count should say that he's not actually good at counting? :P (Though if we did that, I'd be worried about starting to be too long-winded.)
Or maybe there should just be a parenthetical saying there's a reason for not counting explicitly, which we'll come back later. And then we'd need to come back to that when we introduce "bijection" further down the page.