Firstly it is important to know that a hundred times is often being used as just an idiom to mean many times. In the video that is how it is being used, simply emphasising that she called many times.
However, the reason for the a is because it stands for 100, so a single hundred. But because a hundred is also a plural that is why we use times and not time.
Ex: I've tried to solve this a hundred times already.
Ex: She told me a hundred times, but I always forget.