Burong mangga is what Filipinos refer to as fermented green mangoes in the Philippines. It's simply a matter of pickling the fruit in salt.
There's no single method for making buro (=pickle) but here is an easy recipe that you can consult as a guide.
6 pieces of green mangoes (hilaw na mangga) with very firm flesh
half cup of white rock salt
2 cups of water
1. Wash the mangoes thoroughly in running tap water.
2. Peel and cut each mango into three flat pieces -- the middle piece would be the large, flat seed.
3. Shave the flesh off the side of the seed (if you do this skillfully, you'll end up with two long, fat slivers off each mango's seed).
4. Slice up the remaining flesh into long, fat slivers.
5. Arrange the fat slivers upright in a large jar.
Meanwhile, prepare the brine solution by boiling the water and the salt for five minutes. Let cool.
6. Pour the brine mixture into the jar.
7. Cover the jar loosely and let it ferment.
8. After three days, tighten the cap and place the jar in the refrigerator.
To get different flavors, experiment with the brine solution. Add sugar or, for color, small chili peppers that Filipinos call sili.