This animation is likely a bunch of different colored sprites moving or fading in to follow the path of the trail. Look up particle systems. They let you control many sprites/objects at once easily with a set of rules to define their movement.
For this example, you can start with a white circle moving in pre-defined path. Using trigonometric functions is a good way to easily make some curved paths, and you can use bezier curves or splines, although they are more complex to define mathematically. You can also just make it follow the mouse cursor.
Either way you choose, once you have the moving circle, you make a list of sprites to store them in different sizes and colors depending on their order in the structure. You pass on the location to the other sprites in the list the on each frame to create a motion lag, which makes the trail.
This tutorial does a great job at showing how to make particles with this kind of behavior in XNA (MonoGame follows almost the same syntax).