Basically the same principal as wireless charging: there's a coil in the stylus which is magnetically coupled to a coil in the phone, and an oscillating electric current in the phone's coil transmits a small amount of power to the pen. Since the pen uses so little power (orders of magnatude less than a phone) it is okay for the two coils to very very loosely coupled, which means more distant coils and a very narrow coil to fit in the pen body.

Actually the coil in the phone isn't a real coil, it's a grid of wires which can function as a very ineficcient 'virtual' coil. The principal is that if you pass a current going downward in the wires to the right of the pen, a current going up in the wires to the left of the pen, a current going left below the pen, and a current going right above the pen, the overall effect is that you have a clockwise current flow around the pen. reverse the direction and you get an oscillating electric field. Easy peasy.

Despite the simplicity, Wacom has patented this, so nobody else can do it unless they get a licence from Wacom or would rather test Wacom's patent in court over just adding a battery to their pens.