As a native speaker, I have not heared this phrase often, but similar phrases in similar contexts.
In Germany, as in many other countries, it's considered rude to burp loudly. And obviously, you dont get your head chopped off when you do. But its a usual thing to address the rudeness of others with some form of "jovial", yet criticizing remark. Mostly just jokingly criticizing, even though you dont mind the behaviour, but you know its considered rude.
Some examples would be, when you sneeze loudly, some people might yell out "aufwischen!" (to mop up), as if to say, "you sneezed so damn hard you left a puddle".
When people stare with their mouth open, a common remark is to say something along the lines of "mach den mund zu es zieht" (close your mouth, you're creating a draft).
When people had their pants zipper open, people would look them in the eye and just, seemingly randomly, start counting. Everybody understood what that meant. It was kind of an inside joke. The zipper on ones pants is referred to as the "cows barn" and counting would mean "counting down how many cows are escaping", as the "barn is open".
In todays use of my mothers tounge, Aramaic, we have a phrase when someone sneezes really loud. Which roughly translates to "Damn, you woke the dead!"
Now, with the "Stück brot dazu", its a similar remark, but its explanation is a little more disgusting. Bread is served as an addition to many dishes in many different cultures. And in restaurants you can ask for extra bread to your meal. The person making the exaggerated remark jokingly suggests that some of your meal might have come up with that burp and its so massive that it itself counts as a meal. Or that you "re-eat" your food. Therefor they ask you, if you'd like to have some bread to enjoy your "newly created" meal.
Its a friendly little jab, usually not taken as a serious remark but rather playful banter. A sort of inside joke or hyperbole of the other persons behaviour.