the airbags in your car are controlled by sensors that measure deceleration/acceleration and are deployed at some threshold where your head and upper body would likely otherwise impact the steering wheel or side window.

since the deer didn't cause the car to decelerate dramatically, no airbags were deployed. which is a good thing, because it's not fun to operate a vehicle when you can't see. if you crash the front of your car into a large deer or perhaps an elephant, the front airbags could deploy due to the rapid deceleration. if a T-Rex charges through an intersection and T-bones you, chances are your side airbags will deploy due to the sudden lateral acceleration caused by the impact. if you get rear ended by a black rhino at 25 mph, the airbags shouldn't deploy at all since your body should only impact the back of the seat and head restraint.