I know more about the sounds we produce when we speak normally than when we laugh, but I will have a go at answering your question.
When we speak we push small amounts of air out from our lungs which pass over slightly open vocal folds (vocal chords) making them vibrate like strings on a guitar. The sound is then changed again based on the shape of our mouths, lips and the position of the tongue in the mouth.
When we laugh and I mean a real ‘belly laugh’ we often keep pushing air out our lungs until there isn’t much left and we have to take another deep breath. This air passes over the vocal folds the same as for speech, but the vocal folds periodically fully open and close. This makes a rhythmic burst of noise like the ha-ha-ha we call laughing. The shape of our mouths, lips and the position of the tongue will also change the sound.