No, this is absolutely not the first (correct) proof that infinitely many prime numbers come in pairs! That is the twin prime conjecture, and at the time of writing this comment, steadily remains a conjecture. I wonder why Nature, a publication held in high regard, would publish an article with such a title. This is not to say that the result is anything less than incredible.
Terence Tao delivered a very good lecture on this result here [0]