Entropy always increases in closed, isolated systems.
When you are looking at only the formed ice , yes a crystal structure has lower entropy than the same atoms as steam, but when the atoms fell into the energy levels that formed the crystals infrared photons were released as radiation and they also should be counted into the system. The air is heated by them,and entropy overall increases. Precipitation, i.e. change to solid from liquid or gas, always releases heat.
In terms of disorder, one is going to the statistical definition of entropy and counting the possible microstates of a closed system ( air,ice, radiation) will give a much larger number than what (steam) was before precipitation.
Edit for request of "high school level physics".
In my highschool days (1952-1958 gymnasium) the word was not even heard, let alone taught :).
My teacher told me that when super cooled steam suddenly freezes to water the entropy of the system (steam) increases.Why is this true? What I think is that since when steam cools down heat is given out by the system. So the entropy should decrease.
When supercooled steam freezes, it becomes ice. Ice is made of ordered crystals .
Entropy is connected macroscopically with heat changes, and microscopically with the number of ways the atoms and molecules can "move" and reach different configurations. The kinetic energy of the atoms and molecules is what results in the measure of heat, macroscopically.
So the entropy of the icicles is less than when the atoms inside them were free in the supercooled steam.
Also when something cools down the body becomes more ordered so why should entropy increase as my teacher says?
Yes entropy decreases for the icicles, but the rest of the steam, which is the environment in which the icicles are being generated, absorbes the heat and raises its entropy to a higher disorder than when all its molecules were steam.
The crucial thing to remember is that the law of increase of entropy due to the second law of thermodynamics holds for isolated systems, in this case a container full of supercooled steam. You cannot look at the order of the icicles and ignore the increased disorder of the steam.
Now it can happen that all of the particles of steam turn to ice. Then the heat has left as infrared radiation hitting the container walls which have to be taken into account in the entropy balance for a closed system. In a real experiment the steam will be in the atmosphere so there will be atmospheric air that again will take the excess of entropy and the increase in the overall disorder due to the heat transfers .