Aren't sniper rifles effective at that distance (if they hit)? A weapon not constrained to human limits in dynamic movement, heat, sound, pressure, size, etc, isn't effective at a much greater distance?
It's a question of hitting the target with a usefully high probability. At longer ranges dispersion is higher, so even if the aim is correct there's a chance it won't hit. And think of how it works; it shoots out a stream of shells, and tracks the target and the shells with radar. But what if the target is doing some evasive maneuvers, which AFAIU many anti-ship missiles do in their final phases? The shells fly out at about 900 m/s, and a subsonic missile flies at maybe 300 m/s. So by the time the shells whizz by the missile it has moved a quite considerable distance, and while the radar tracking can apply the correct amount of lead (with a Kalman filter or similar to take into account measurement noise), if the missile is doing some unpredictable maneuvers that might not help. With the missile closing at 300 m/s, you don't have very many cycles of shoot-see-where-the-shell-goes-adjust-aim before the missile slams into the ship.
And to make matters worse, there are also supersonic anti-ship missiles. Now evasive maneuvers in the high-density air at the surface isn't really a concept that works out for Mach 3 missiles, but, well, the missile compensates by flying as fast as the cannon shells.
Hence stuff like this RAM thingy is seen as a better system against high-end anti-ship missiles.
These are ballistic, like bullets or shells, but have some capability for maneuver, as with a guided missile. A smart munition with longer range might make for a more effective countermeasure against high-speed (supersonic or hypersonic) missiles. Or possibly high-speed (supersonic) torpedoes, another emerging class of anti-ship weapons.
The effective range isn't limited by the ballistics of the weapon (The 20 mm shells it fires will easily go 5+km), it's limited by actually hitting the damn thing you're aiming at.
Assuming your firing at a missile flying right at you, and assuming a Cold War era threat (so say a Kh-22 fired from a Tu-22M), you're talking about a 92cm diameter target (46cm radius) that you're trying to hit.
In reality, things are worse because in addition to dispersion, your actual point of aim of your stream of bullets will be effected by external factors (like wind).
You can see how in the scenario layed out above, you do actually have what seems to be a reasonable chance of landing a few hits when you're spewing out hundreds of rounds within seconds. But you can also see how quickly the odds go to crap as you push the range further and further.
I think that gets to the reality of what the "effective range" value is. It is a rule of thumb designed to help weapons officers quickly determine how best to react in a scenario.
Remember that CIWS was envisioned to be deployed in a scenario where USN ships would be subject to saturation attacks by supersonic anti-ship missiles from multiple bearings and sources. Phalanx itself only has a magazine of ~1000 rounds, which is good for maybe only 10s of firing, so you really do need to conserve ammo.
What happened to "Metal storm"?
1: https://old.reddit.com/r/NonCredibleDefense/comments/16xhmpv...