I am an ATCO working in India. I am trying to understand why an aircraft's rate of descent increases, when its speed increases. I am particularly interested in how it relates to jet engine aircraft.
Is it the same for aircraft climb rate?
If the pilot commands a certain glide path angle, any change in airspeed will also change vertical speed, because both are connected by the glide path angle $\gamma$.
Note that the vertical speed is air speed$\cdot\sin\gamma$. Climbing or descending makes no difference; only the sign of $\gamma$ changes (positive $\gamma$ means the aircraft climbs). Also the type of propulsion will not affect this basic relationship.
Since descending reduces potential energy, some other energy will increase. If the aircraft speeds up, its kinetic energy increases until the increase in drag balances the energy gain. Reducing engine thrust or deploying spoilers will limit the speed increase, but for a steep descent, even with idling engines and full spoiler deflection the flight speed will increase.
Is it the same for aircraft climb rate?
No. In a climb the potential energy must be increased, and the primary source of energy are the engines. Only now will the type of propulsion affect the result: Both the speeds for the steepest and the fastest climb for propeller aircraft are fairly low, so an increase in vertical speed will normally be accompanied by a reduction in flight speed, freeing up more excess energy for climbing.
Jets also need to fly slow for their steepest climb, but need to pick up speed for their fastest climb speed. This is the more pronounced the lower the bypass ratio of the engines is, so fighter jets need speed to climb quickly. Turbofan-powered airliners fall between propeller aircraft and fighter jets and prefer a moderate speed for their best climb performance.
This is all only true in stationary flight. In a pull-up, any aircraft can gain altitude quickly by trading kinetic for potential energy. But this is a transitionary effect - the high climb speed will only last while the aircraft has excess speed to spend.
The other answers handle the question very basically. If a plane is moving through the air in the direction of its nose, and that nose happens to be pointed toward the ground, then the faster the plane is moving through the air in the direction of its nose, the faster it will descend. This is simple trigonometry; the plane's "true airspeed" is the hypotenuse of a triangle, the other two legs being the pure horizontal and vertical components of that airspeed, namely the plane's "ground speed" and its "vertical speed" (aka climb/descent rate). At a constant angle of climb or descent, if you increase your airspeed, you will increase your ground speed and your vertical speed.
Another, more realistic way to think of it is the other way around; all other things being equal, when an aircraft's descent rate increases, its speed increases. This is due to physics, and the conversion of energy between potential and kinetic forms.
A simple non-aircraft example. Say you have a sandbag with a rope tied around it and fed through a pulley overhead. By pulling on the rope, you lift the sandbag against the force of gravity by a certain amount. You expend energy to do this, and are storing some of that energy (ideally, all of it, but in the real world there are "inefficiencies") in the weight inherent in its height above the ground. Release the weight, and gravity accelerates it by a time-dependent (and thus height-dependent) amount, so that by the time it hits the ground, all the potential energy you stored by hoisting the weight has been converted to kinetic energy of motion.
This same basic relationship also exists in aircraft; an aircraft stores energy expended by its engine (or in a glider, by the engine of a tow craft) in its altitude. The aircraft can then reduce its altitude to maintain (or in a true dive, even gain) forward airspeed. This is one of the most basic principles of aerobatics and is taught to all pilots; you can "trade altitude for airspeed", whenever you need more (or less) of one and have enough of the other to give up (or gain).
The mechanisms by which this happens can be made as complex and esoteric as you want to get; we can cover the four fundamental forces of flight and how different forces act in different directions depending on the attitude of the aircraft and thus contribute different horizontal and vertical subcomponents to the aircraft's motion relative to its nose or the ground under it. But, to answer this question it is enough simply to state that as a plane descends, it is trading its altitude to gain energy, and the amount of that energy not lost to drag will increase the plane's forward airspeed.
This feeds back into the original answer. As you descend from altitude, your forward airspeed (the speed of travel in the direction of the plane's nose) will increase as potential energy is converted to kinetic. Because that forward airspeed is in a downward direction, as long as drag is not keeping acceleration in check (such as with a higher angle of attack, which increases drag by increasing the "cross section" of the wings directly exposed to airflow), it creates a feedback loop; you descend, gaining airspeed, which causes you to descend faster, gaining more airspeed.
To counteract this, descending airliners often reduce engine power (so the energy gained by descending merely offsets the decrease in thrust), increase their angle of attack (if the aircraft is travelling along a path "beneath" its nose, the wings are more "side-on" to the airflow moving past them which increases drag) and deploy spoilers or air brakes (which directly increase drag even at low angles of attack).
So unlike the answers that were here first, I say that the short version is that "out in the real world", it doesn't work that way.... except in some situations that are commonly seen as an air traffic controller.
When a pilot is given a descent, with no additional instructions, we will typically descend either at a constant rate of descent or a constant airspeed. The AIM says that we should descend at an optimum rate (not the maximum rate) until within 1000 feet of the assigned altitude, and then no more than 1500 fpm for that last 1000 ft. Depending on my distance from the destination or other expected crossing restriction, I typically choose a descent rate between 1500 and 2500 fpm and use power to maintain my target descent airspeed. When using this method, the descent rate will not increase with speed, however my vertical flight path angle will change as the speed changes.
The behavior that you describe is most likely seen when a crossing restriction is issued to the pilot, or they are planning to cross a self-made crossing restriction (10,000 ft. AGL at 30nm from the airport is commonly used). This effectively fixes the vertical flight path angle, and now the only way to compensate for speed changes is to adjust the vertical speed.
When descending from high altitudes (above 30,000ish feet, depending on the airplane), the indicated airspeed will increase with a constant Mach number, until the target descent airspeed is reached, and vertical speed must be increased in order to compensate for the higher ground speed in order to make the crossing restriction. On the other hand, if a speed reduction is assigned then the descent rate may be reduced in order to make the same crossing restriction (or the aircraft just reaches it's assigned altitude early, which is not normally an issue).
This adjustment happens either automatically (by the FMS / autopilot in VNAV mode), or is monitored and manually adjusted by the pilot in order to meet the crossing restrictions.