Why prohibit engine braking?

Even if the signs are often referring to engine braking in general, the intention is to prohibit usage of so called compression release engine brakes. Diesel engines in heavy vehicles are often equiped with such a braking system, which is more effective than just braking with an idle engine, but can be very loud. The purpose of prohibiting such braking systems is to avoid noise pollution in built-up areas.

I was always told that this is because engine braking is noisy[1], and hence undesirable in built-up areas..

In New Zealand there are signs "Heavy Vehicles Please NO Engine Brakes NEXT 4 km", as these vehicles will be much louder than typical cars.

[1] https://www.nzta.govt.nz/assets/Highways-Information-Portal/Technical-disciplines/Noise-and-vibration/Research-and-information/Leaflets/Engine-Braking-noise-leaflet-v1.1.pdf

These signs are usually directed at large trucks, which make a lot of noise when engine braking. Example. So it's intended to reduce noise pollution. The rule may or may not apply to smaller vehicles.

Ok, so there's alot of misconception here... I am a Professional truck driver. I'll mention the misconceptions one at a time:

1) engine (Jake) brake laws are antiquated. New big rigs (roughly 2007 (maybe a smidgen earlier) to current date) are inherently so silent when engine braking, you really can't tell there is engine braking at all. These laws are for older trucks that have been grandfathered to not require mufflers. The laws were to combat noise pollution. Big trucks also down shift while using the engine brake itself to slow down, and in a lot of cases, rarely touch the brake pedal when done correctly. Even without the actual device itself, we still refer to it as engine braking because we literally use the engine to slow down when we don't need the engine brake [device]. The officer must prove that you were actually using a Jake brake.

2) Double clutching has nothing to do with engine brakes nor slowing the truck down. You can actually shift a big rig without EVER touching the clutch pedal. It is called "Floating Gears". You SHOULDN'T float gears in a car, but it isn't impossible. I do it often in a car. In NORMAL passenger cars, you don't double clutch because they have synchronizers that prevent the need of it. You can tear up your transmission by double clutching a car [that has synchronizers].

3) Cars don't have engine brakes. They don't need em. And VERY FEW diesel pickups have them, cause they too don't need them.

4) Someone mentioned dynamic braking using the cooling system and it [dynamic braking] has absolutely nothing to do with the cooling system. It's primarily used in trains to help the locomotives slow down... Dynamic braking is used in electric motors. Not internal combustion engines. It's the high to low RPM's with no throttle and the car's low weight after shifting that slows the car down (the motor wants to idle, so the low weight of the car causes a much more rapid deceleration than a big rig, where the mass of the big rig wants to keep the rig going, and over-rides the engine's wish to idle).

5) all of the terms in reference to the Jake brake are synonymous... Jake brake, engine brake, compression brake, etc are all the same thing.

EDIT: Please stop down-voting my answer and editing it because you don't THINK I wrote it correctly. I wrote the answer how I wanted it to be written. Thanks.

To succinctly answer the actual question -- assuming the reason for the signs in the USA is the same for the identical signs here in AUS:

As answered by other people, it is to reduce noise (excessive noise).

To further flesh out my answer a bit...

As I understand it, at least here in Australia, there are at least three different types of braking:

'Wheel' Braking The quietest but possibly the least effective, especially on slopes, due to overheating and brake fade, etc.

Exhaust Braking [To me] has quite a pleasant 'Brrrr' sound. Not silent, but not all that noisy either.

Engine Braking (Jake Braking) The most effective, but also, by far, the noisiest.

Links: Brake Fade (Wikipedia) (See esp. 'Causes of Brake Fade' section) Jake Brake vs Exhaust Brake (Difference Between)