In general, heating causes a fluid to expand in volume, reducing its density and causing it to rise.
At the beginning of this experiment, because you thoroughly mixed the soapy liquid in the pan before heating it, the temperature and density of the soapy liquid is the same throughout. When the pan is heated, the liquid at the bottom of the pan heats up first, becoming less dense. This lighter liquid rises in localized columns. When this warmer, less-dense fluid reaches the surface, it cools, becomes heavier, and sinks.
The localized region where the fluid rises and sinks is called a convection cell. Adding food coloring highlights the movement of these convection currents.
When the pan is placed on the hot plate, convection cells rise through the liquid directly above the heating element (or coins, if you used them).
When the warm pan is then placed on a cool surface, the convection process slows. This causes the convection cells to widen and become extremely well defined. This slowing of the convection process allows the solution to calm down enough for darker, curved lines to appear. These dark lines show where the fluid of one or more convection cells is diving downward.