Why did the Ju-87 Stuka have a siren? Was this for purely psychological reasons or did it help the pilot in some way?
The B-1 [variant] was also fitted with "Jericho trumpets", essentially propeller-driven sirens with a diameter of 0.7 m (2.3 ft) mounted on the wing's leading edge directly forward of the landing gear, or on the front edge of the fixed main gear fairing. This was used to weaken enemy morale and enhance the intimidation of dive-bombing. After the enemy became used to it, however, they were withdrawn. The devices caused a loss of some 20–25 km/h (10-20 mph) through drag. Instead, some bombs were fitted with whistles on the fin to produce the noise after release.
So, yes, it was purely psychological and actually hindered the pilot by reducing air speed in ordinary flight and they were removed when they stopped having the psychological effect.
The psychology behind it helped the pilot ... panicking enemies doesn't make for very good shots, they're more likely to miss you. They're also more likely to just drop flat on the ground rather than dive into cover or try to shoot back at you.
And that was pretty much the idea. Get the enemies to become disoriented, panicky, so they're less efficient fighters and easier to defeat by both you and the ground forces you're supporting.
Another reason for the siren is that during the first years of WWII most army's ground transportation was horse drawn. You freak the horses and the unit does not move. The French, Polish, and Russian armies used primarily horse drawn transport. The British and American armies used trucks. A truck can't freak out like a horse.