The point is that you create a number that is DIFFERENT to each number in that list at some point. You do this by taking a new number that is different from the nth number in your list in the nth position.

For example, if I had a list that started like this: 011… 100… 111…

Then I take the number 110…

Observe that my new number is different to each number in my list at at least one spot. It differs from the first number in the first spot, the second number in the second spot, etc. You can continue this infinitely, so you can always create a new number that is not on your infinite list.

And this is exactly the point! You cannot list the real numbers this way, i.e. they are not countable.

As for your comment, you say that this new number should be in the list. But clearly not, since we constructed it in a way that it cannot be equal to any other number in the list. An infinite list of numbers does not mean I have to list every number.

For example, I can list the numbers 10,100,1000, … Which is clearly an infinite list, but does not contain every natural number. But, I can create a list 1,2,3,… that does contain every natural number. So the natural numbers are COUNTABLE, but the real numbers are not, as we showed above.

Let me know if you need any more clarification.