From what I understand, in all of the R-CNN family of models (R-CNN, Fast R-CNN, and Faster R-CNN) there is a regression head that specifies how the bounding box should be modified from the proposed region. But the regression head is not given the information of the entire image, it's only given the part of the feature map that corresponds to the proposed section. If this proposed section isn't large enough to contain all of the object, how is the regression head supposed to know how much bigger the bounding box needs to be?
Usually at least one of the anchor boxes is large enough to contain most if not all of the object. Also note that each neuron at the end of a backbone network has a pretty large receptive field, so the actual area which is "seen" is even larger than the anchor box. Finally, if all else fails, the network will simply have to guess the full extents based seeing a significant fraction of the object, which is usually not too difficult either.