It implies a strong, uncontrollable laughter that is often loud and spontaneous.
Here are some examples of how it's used:
* Me partí de risa con tu chiste. (I burst out laughing at your joke.)
* No pude evitar partirme de risa al ver su cara. (I couldn't help but split my sides laughing when I saw his face.)
* Se partieron de risa cuando escucharon la historia. (They burst out laughing when they heard the story.)