Here’s video of Phidippus adumbrata jumping spiders mating.
The male has two primary concerns: he wants to mate, but he doesn’t want to get eaten. His elaborate dance is not only gauging interest, but possibly also determining his risk level in approaching the female.
Jumping spiders, in general, are very visually-oriented creatures. They have excellent vision in color. According to some studies, the arm-waving behavior is not purely visual, however, but also vibrational, and the dance differs if the male is approaching the female in her nest or out in the open.
Anyway, spoiler alert, this male succeeded in mating and avoided being eaten (at least for the time being).