The objective of the game is to build bigger turtles by combining the smaller turtles. To do this, you have to match the colors of the circles (which are chosen on the right side of the UI) with the color arrangement of the 2x2 squares of turtles (left side of the UI).
For example, if in a turtle square all four turtles are of different colors, then you have to select the last arrangement (all four circles are of different colors) and then click on those four turtles, this will combine them into a purple turtle. If instead all four turtles have the same color, then you have to choose the first arrangement (all four circles have the same color) and then click on those four turtles, this will combine them into a red turtle.
This applies for all squares of turtles. Some arrangements have to be rotated to work though, which can be done with the A and D keys; with the middle and right mouse buttons; or by clicking on the buttons with rotating arrows on the screen.
This explanation is a bit wordy. Hope it helps!