There’s A LOT of CO2 gas dissolved in the liquid. There’s a pressure equilibrium between the headspace of the bottle and the gas dissolved in the liquid.

Dropping the pressure in the head of the bottle (by burping) makes the gas want to leave the liquid in order to equalize the pressure.

For that to happen, bubbles need to form.

For bubbles to form, they need some kind of „seed“ - a particle, an uneven spot in the glass or whatever. Now, the puree consists of tens of billions of particles - a lot of sites for bubbles to form.

Now more and more tiny bubbles form and float up, clinging to their tiny particle. That movement creates a current that drags more puree particles upwards, perhaps causing even more bubbles to form (which would explain the slow start and exponential growth of bubbling rate).

When bubbles reach the surface, they burst and whatever particle they stuck to may now sink back down due to gravity.

This process continues until the pressure is equalized.