The theory is simple. You show the spider a screen with a moving shape that differs from the background in color but not in brightness. And you see if the spider tries to follow.
The problem with letting the jumping spider actually turn and respond is that they'll absolutely do so, but it changes some of what they see. So, what we want is to really have some control over what the spiders can see at any given moment. So the researchers hold them in place with tiny magnets attached to their heads.
What we do is we give them a ball to stand on. They actually hold it with their feet. And we can monitor how that ball moves around in their feet to know where they would want to go.
If the spider turns the ball to the left, it's probably trying to look to the right to follow the moving shape. And that's evidence the spider can discriminate between the colors of the shape and the background.