At training time, the probability for a given location is 1.0 if an object is present and 0.0 if there is not. The output of the CNN includes a value for object presence. The predicted value is compared to the ground truth value in the loss function, and hopefully iteratively approaches it.
At runtime, the CNN outputs that object presence value given the input signal and its learned weights. The value is constrained to be between 0.0 \leq x \leq 1.0 so that it can be treated as a confidence or probability. [know any activation function that would be helpful???]
There is an example of setting up the training data that touches on this in one of the replies in this Discourse thread: Quick question regarding YOLO algorithm - #3 by ai_curious