How is Doppler radar used in rain prediction?
The easiest answer is that Safe Haven is a movie, and the writer made a factual error (which is not uncommon). I have not watched that movie, so I am unsure what type of situation they were dealing with. Doppler radars do not predict, but they observe.

What is possible, however, is extrapolation and the creation of inferences. For example, tornadoes are not directly predicted by Doppler Radar, but by making inferences from the data, a meteorologist may detect a tornado. In a similar sense, if the radar-derived 'storm total precipitation' has estimated an average 0.25 inches of rain from a squall line for the past 200 miles, one may extrapolate, or estimate, that the squall line will produce ~0.25 inches of rain. Could the squall line alter its path or intensity and drastically change the amount? Sure, but without additional information, including information not derived from a radar, that would be a difficult task.

Edit: Ok, I think I have a better idea of what the question was and how it can be answered.
Doppler radar can measure wind speed relative to the radar site. So if can sense how fast a storm is approaching. Assuming the storm does not change it's speed, and given it's history, you can infer when it will arrive at the radar site. For example, if a storm is 50 miles away, and it moves at 25 miles per hour it will arrive in $50\text{ mi}\div 25\frac{\text{mi}}{\text{hr}}=2\text{ hr}$, provided the storm does not change.