First consider a reference path. Assign an arbitrary phase to it. This will be our reference phase. Since phase can be represented as a unit vector in the complex plane, this will be represented by an arrow. Now consider the phase for a nearby path, this will be another arrow. So on for all the paths. To get the net phase, you need to add all of these vectors. And to add vectors, you place the tail of one to the head of the other. This is what the animation shows.
Here is an example image, shown in Feynman’s highly readable QED book. It is talking about a light emitted at $S$ and detected at $P$. $A-M$ is a mirror. Each path has an associated time (taken to cover the path) with it which tells us what the phase is. So we pick a fight pointing arrow to represent our reference phase and calculate the relative phases accordingly.
We then add all the phases to get the total probability (related directly to phase) and this is done by placing the arrows tail to head. In other words, a vector addition.