I could see why this might be a difficult phrase to find in the dictionary. (I had to check a few dictionaries before I could find a definition that matched this context.)
I did manage to scrounge this one on Wordnik:
breakout (verb transitive) (idiomatic) to bring out, use, or present
So, the original sentence could be rephrased as:
My husband, notoriously bad for forgetting to add things to the calendar because it was too much of a hassle to take his phone out of his pocket and open the app.
The Free Dictionary has a similar definition:
breakout take from stowage in preparation for use
So, “stowage” would be where the phone is normally kept, which in this case would be the husband’s pocket.
Another way this could have been worded is by using the phrasal verb whip out (although that implies the movement is sudden, which may not be the case here):
whip out (verb transitive) to pull something out with a sudden jerk