It suggests an intense feeling of inquisitiveness that is almost impossible to contain.
Here are some examples:
* "The children burst with curiosity when they saw the magician pull a rabbit out of his hat."
* "She burst with curiosity about the contents of the mysterious box."
* "The journalist burst with curiosity about the politician's secret."
The idiom emphasizes the overwhelming nature of the curiosity, implying a strong urge to discover and understand something new.