OP's context is an example of OED's definition #3 for the phrasal verb to call out (first recorded 1823)...
To challenge to fight (esp. a duel).
It's effectively a figurative extension of usages such as...
"Come outside and say that!" (repeat your insult, and we will fight where it is more convenient)
...but the modern usage doesn't directly refer to fighting with duelling swords or fists. The metaphoric "weapons" which will be used in the "contest" are justifications (facts and logical arguments).
By implication, the person issuing the challenge expects to win, because he doesn't think the other person actually has valid justification for whatever he said, implied, or did. It's important to note that you can just as easily be "called out" over disapproved-of action (or inaction) as a disputed statement...
"If your husband never helps with the housework, you should call him out about it."
That's to say, to call [someone] out over/about X means to demand that they justify X.
There is an important difference not explicitly mentioned in those definitions: the grammatical subject of break off refers to a different agent than the grammatical subject of break up.
The subject of break off was previously participating in or causing the activity that ended. The activity ended because the subject stopped continuing it.
The subject of break up was outside the activity, not part of it.* The subject actively interfered with its continuation, causing it to stop.
For example:
The police broke up the fight.
means that some people, not the police, were fighting, and the police forcibly made them stop fighting. Most likely, the police physically grabbed the combatants, pulled them apart, and restrained them.
The police broke off the fight.
means that the police were themselves involved in the fight, and stopped fighting by their own choice. Perhaps the police were defending a building against a crowd during a public protest, and the police lost heart and decided to walk away and let the crowd do as it pleased. Or perhaps the "fight" was a negotiation to get a raise in pay. Then if the police "broke off" this fight, that would mean that they gave up trying to get the raise, agreeing to continue to work at their present salary.
When people "break off negotiations", this means that they refuse to continue negotiating. They have given up hope of making a deal, or they demand a concession from the other side before they will even continue talking. They have "walked away from the negotiation table."
*Of course, there is also a different sense of
break up, such as
sense 2, the break-up of a romantic relationship, in which the participants do end it.
Best Answer
According to the Cambridge Dictionary
bubble
In the previous paragraph of the referred article such bubble is described
Breaking out of the bubble is obtaining podcasts' suggestions from outside your circle (friends in Facebook, people you follow in Twitter, blogs you usually read, ...)
Notice that bubble is not a synonym for any problem. It's a specific situation whose details are explained in the dictionary entry. Breaking out the bubble does not mean solving any problem that you may face, that situation may not be even a problem for you. I've described what that expression means in your context.