The performer displays a box with a see through front with bars across, - the type of box one would normally use to carry a pet. The box seems to be clearly empty. A balloon is placed in the box, which can easily be seen through front bars. The balloon is burstwith a large needle, and instantly transforms into a Bunny rabbit.
You could use a pair of doves or other small live stock (white mice, baby chicks, etc.) in lieu of the bunny or large number of inanimate production items like silks and flowers, sponge and rubber items, etc.