He probably thinks that in order to add (1) to an infinite sum of (1)'s you have to actually reach the end of that infinite sum and place it precisely at that point.
The fact is, you can add that (1) at any position in the sum. Why are these people obsessing over the non-existent end of infinite sums?
Of course, there is no last position, so you can’t add it at the end of the sum. But you can add it anywhere else. It makes no difference.
I suppose the problem is that, since all of the terms are equal, the standard way of representing infinite sums makes the result of (\infty + 1) look the same as (\infty).
What do you get when you take (1 + 1 + 1 + \cdots) and add (1) to it? Well, you get (1 + 1 + 1 + \cdots). The symbols are exactly the same but they are representing different things.
One way to solve this problem is to maintain the identity of terms. So instead of writing (1 + 1 + 1 + \cdots), we could say (One_1 + One_2 + One_3 + \cdots) so that when we add (One_{new}) to this sum, we could represent the result as (One_{new} + One_1 + One_2 + One_3 + \cdots).
But since we’re comparing sums that are infinite, even if we do this, it would still be very easy to pretend that they are not different.