According to the Cambridge Dictionary
a situation in which you only experience things that you expect or find easy to deal with, for example opinions you agree with, or people who are similar to you
In the previous paragraph of the referred article such bubble is described
many of these routes take us to more of the same thing, or the same
shade of different. The politics and the aesthetic tend to fall within
a narrow — and comfortable — part of the preference spectrum.
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.