In experiment, cold skillet with droplets of water was filled with layer of oil and heated. The droplets started exploding. Why water can not leave skillet less violently, like steady chain of steam bubbles ?
The bubbles explode because they superheat. The oil prevents the nucleation of steam bubbles you get when heating water in a pan, and the water drops turn to steam all at once and of course the steam expands explosively.
Even though water and steam are in equilbrium at 100°C there is a potential barrier to formation of a steam bubble. This is because small bubbles have a high pressure (the pressure is inversely proportional to the bubble radius) so you need a lot of energy to start a bubble. When you heat water in a pan you'll find steam bubbles nucleate at defects (e.g. scratches) in the pan surface. You'll often see streams of bubbles coming from the same point.
In your experiment the oil probably dewets the water droplets from the skillet so you have a water drop surrounded by oil and there is nothing to act as a nucleus. If you heat the oil slowly the water drops will superheat until eventually the temperature is high enough to nucleate bubbles. At that point part of the drop will turn into steam almost instantly and then expand explosively. You can work out how much of the drop vaporises by multiplying the superheating (i.e. the temperature - 100°C) by the specific heat of water, and then dividing this energy by the latent heat of vaporisation. As I recall, only a relatively small part of the drop vaporises, but of course this generates a large volume of steam
Even in ideal circumstances there is a limit to how far you can superheat the water. At some point the superheating is so great that nucleation will be triggered just by random fluctuations in the water. You couldn't superheat indefinitely even in perfectly pure oil and water in zero G. Obviously in your experiment the bubbles nucleate a lot earlier than this.
You get the same effect heating distilled water in a microwave oven. If you use pure water and a clean glass bowl there is nothing to act as a nucleus for the steam bubbles and the water can superheat. Sadly you often discover this as you remove the bowl as the vibration can induce nucleation and an explosion of steam - hopefully not in your face!
Because the water is covered by oil (oil is up because its density is just 90% of the density of water) and the oil has no holes in it! The oil has no holes because of the surface tension, a term in the energy that punishes too high surface of the oil; for the same reason, the oil resists the creation of new holes through which the steam could escape or any significant non-uniformities of the oil-water boundary, for that matter. So the steam grows as a bubble beneath the oil and pushes the whole body of oil it is covered by, ultimately giving the oil a sufficient velocity so that it shoots and makes the stove a bit more messy again.