Hold the ocarina in playing position: whistle end in your mouth, first two fingers of your left hand covering the left-side holes, first two right fingers over the right-hand holes.
Compare the position of your fingers to the chart. Place your fingers in the DO position (top left). The four black circles indicate that all four holes are fully covered.
Lift the middle finger of your right hand and blow. You are playing RE.
Imagine that the top of the ocarina is where the name of the note is on the chart. When you look down at your fingers, the ones closest to you are the bottom row in the diagram. The ones farther away are the top row. Still looking at RE, the open circle shows that your right middle finger is lifted off the hole.
Start over at DO. Follow the chart from top left to bottom right, lifting the appropriate finger whenever you see an open circle. You just played a scale. Try the reverse. Repeat until you feel comfortable.
Try the chromatic scale. If a circle is half-filled, cover half of the hole.