Before everyone freaks out, no, you don't use petroleum oil. You use vegetable, fish or animal oil. In earlier times, whale oil would be used. The OP's picture looks like a fuel oil leak, not an attempt at wave calming.
I have seen references of this technique being used since at least the early 1800s, probably much earlier. Ernest Shackleton made use of it in 1916 during the Voyage Of The James Caird across 800 miles of Arctic Ocean to calm the very rough seas.
"The Popular Science Monthly, Volume 43", from May 1893, contains an article "Why A Film Of Oil Can Calm The Sea" by G. W. Littlehales from the US Hydrographic Office (page 494). Here's a description of one schooner captain's apparatus and its effectiveness in heavy storm.
The sails were blown away, men washed from the pumps, and the boats and other things above the deck wrecked by the heavy seas... Two wooden, ten-gallon kegs, containing boiled linseed [flax] oil, were lashed to the quarters of the vessel. The oil was allowed to ooze out through two small holes in the heads of the kegs. The effect was all that could be desired. After the oil had spread, no water came on board, the men returned to the pumps, the vessel was pumped out, and the decks were cleaned up. During the sixteen hours in which the oil was used eight gallons were expended.
From that testimony, and the article mentions examining thousands of reports, it seems very effective and economical. Just two quarts per hour of thick, heavy oil...
...when allowed to drop into the sea soon spreads over its surface, forming an oily layer within the area of which the waves, instead of breaking, become huge rollers upon which the vessels rise and fall without shocks and without shipping any water.
The oil doesn't prevent the waves, but it prevents them from breaking and spraying water onto the deck and into the ship.
The article goes on in some detail describing wave action. What it comes down to is the oil acts as a lubricant between the wind and the water. It smooths the surface of the water so the wind has no rough surface to drag on and cause the waves to spray and break.
When wind blows over water, all the air does not pass over the surface of the water. On account of the high degree of adhesion between air and water, a thin stratum of air remains in contact with the water, and it is the action of the internal friction or viscosity of air tending to draw this stratum along which causes the tractive effect of wind on water.
When a film of oil is spread over the surface, this tractive force is not brought to bear on the surface of the water as long as the film remains unbroken... The surface of the water is thus shielded from the action of the wind... the only action of the wind in such a case is to move the film over the surface of the water.