Similarly, why is power available (nearly) constant with speed for a propeller engine, while it varies for a jet engine?
Turboprops and turbojets - or, more broadly, jets - produce thrust in somewhat different ways.
First of all, let's address the way thrust is produced. Per Newton's 2nd and 3rd laws, force equals acceleration times mass, and an action (accelerating the air) produces an opposite reaction. After canceling out the variables (the math is easy to find), thrust is proportional to T=v*m' (m'=mass flow rate), and power transferred to the air is proportional to P=v^2*m'/2. All velocities are in the airplane's frame of reference.
Now let's go to how engines produce this thrust. A jet engine first decelerates the incoming air to a near-zero velocity, generating drag, then accelerates it to a constant velocity, higher than the initial one, producing thrust. Both v and m' for a jet engine vary across the envelope, but they change much slower than the plane's speed. The engine spends roughly the same amount of power per unit thrust at any velocity.
A propeller doesn't decelerate the air at all. It only accelerates the incoming air by some amount. This converts power P into velocity v. Now, per above, P~v^2. If you encounter air moving at 0 m/s, you need 5 kJ/kg to accelerate it to 100 m/s. If the air is already moving at 100 m/s, you need 15 kJ/kg to get it to 200 m/s.
The result is that propellers encounter less resistance at low airspeed, so they get more thrust per horsepower the slower the plane is. This allows them to push more air or push it faster, producing more thrust. For fixed-pitch propellers, this works by producing more air delta-V and less drag. High-performance turboprops tend to have variable pitch propellers, which will be adjusted to push more air (slower) at low airspeed, or less (but faster) at high airspeed.
Turbofans, combining a fixed-pitch ducted propeller with a turbojet core, are somewhat in between. They lose some efficiency and some thrust as they gain more speed, like propellers, but their curve is much smoother and closer to turbojets in this regard.
The above is an extreme simplification (as is the jet's thrust curve in the question), just enough to get the idea across. The jet's actual thrust is also non-linear, which has been addressed in another question: How (and why) does engine thrust changes with airspeed?
Assuming that the net thrust of a turbojet is constant is not correct. It is assumed to be constant (for simplicity by the aircraft performance engineers and usually valid for low subsonic speeds), but in reality, the performance is not constant, and it also varies with altitude. This is best shown by a simple simulation of a turbojet engine. The following graph shows the net thrust as function of the Mach number. It clearly shows that the net thrust is not constant with speed:
The inlet of the turbojet slows the flow down and creates the optimal conditions for producing the exhaust jet. A turboprop relies on adding energy to the freestream air, as the speed of the airframe increases, there is less energy can be added to accelerate the flow.
There are several effects which in combination make constant thrust a good approximation at subsonic speed.
Thrust is created by accelerating a working mass in opposite direction. Net thrust is the difference between the impulse of the air flowing towards the engine and the combined impulse of burnt fuel and the air exiting the engine (and propeller, if one is fitted), derived after the time. Since that impulse is the product of mass and speed, you can either accelerate a large mass by a small speed difference, like a propeller does, or a small mass by a large speed difference, like a turbojet does.
When flying faster, the entry impulse of a propeller quickly grows large relative to the exit impulse, so thrust goes down with the inverse of speed. On the other hand, the high exit speed of a turbojet results only in a small increase of the entry impulse relative to the exit impulse while speed increases.
But if that were all, even the thrust of a turbojet engine would drop when speed increases. But there is a second effect which helps to let thrust grow with speed. With the square of speed, to be precise. That is the ram effect which helps to precompress the air entering the engine. At subsonic speed, this just about compensates for the loss of thrust: At low speed, the growing entry impulse lets thrust drop a bit but at higher subsonic speed the ram effect becomes larger and raises thrust again, such that a constant thrust becomes a good approximation. However, at supersonic speed the ram effect becomes dominant and thrust grows with speed squared – until the absolute internal pressure becomes too high so the engine must be throttled (or the aircraft needs to fly higher) or the shock losses in the intake become too large and thrust drops again.