Objectness is a binary cross entropy loss term over 2 classes (object/not object) associated with each anchor box in the first stage (RPN), and classication loss is normal cross-entropy term over C classes. Both first stage region proposals and second stage bounding boxes are also penalized with a smooth L1 loss term.
It should also be noted that the authors train the first and second stage alternately since both rely on the same features computed with convolutional layers + FPN to aid in training convergence.
Not a very clear description? I'd recommend reading the original Faster-RCNN paper as it is pretty foundational and will probably do a better job describing the loss terms than me.