In the opening stanza, Duffy describes the "recklessness written all over" the love letters that she wrote many years ago. The implication here is that when she was younger, she loved recklessly, perhaps meaning that she loved too intensely or was governed too much by her emotions. One might be reminded here of Romeo and Juliet, who also love too intensely and, arguably, recklessly.
This idea of loving recklessly may also be familiar to many readers of the poem from their own experiences. Indeed, when we are young, we do tend to become obsessed or infatuated with the loved one—and with the feeling of love itself—much more easily than we do as adults. This is because love felt for the first time is perhaps more exhilarating than love felt a second, third, or fourth time. This is a moving idea, because it is at once a joyful feeling to remember our first experiences of love and sad to think that love often doesn't have quite the same impact when we are adults.
In the second stanza, Duffy writes that "Nobody burns" their old love letters. If someone does not burn their old love letters, it is possibly because they want to hold on to those special feelings and sentiments that they experienced when they loved for the first time. The implication is that one might want to hold onto those feelings and sentiments because they are unable to experience them as fully and as intensely as adults. This is a reiteration of the idea discussed above and is moving for the same reason.