To get a mango on a stick, make an incision with a paring knife at the base of the mango just to the side of the fibrous pit. Push the knife in about 1 inch and pull straight out. That will leave a perfect guide for a popsicle stick to be inserted--the perfect sturdiness for a mango. You can then very easily peel it with a "Y" vegetable peeler without handling the sticky, slippery mango.
It's not that complicated. The hardest part is impaling a mango with a sturdy stick. Then, working vertically, slice five "petals" off the outter perimeter, starting a little higher than halfway up the mango. The pieces should be about 3/4" thick for easy eating.
For the next layer in, make your cuts so the center of the next set of petals is where the gap between the first petals is (so that the petals alternate...hard to describe but just look at your photo). Keep working in until you reach the center.