It is not "forgive and forget" as if nothing wrong had ever happened, but "forgive and go forward," building on the mistakes of the past and the energy generated by reconciliation to create a new future.
In any activity that requires concentrated effort, the breath quite naturally plays a role. If you have ever tried to thread a needle or repair a watch, you might have observed that without even thinking about it the breath quiets and deepens. Singers, swimmers, people who struggle with panic attacks, and a host of others learn the importance of proper breathing in order to negotiate the respective tasks at hand. Thus, that the art of contemplative practice can be facilitated by the breath should come as no surprise.