static thrust is the thrust generated by the engine when the plane is at rest with respect to the Earth and the surrounding air. So why would it have different values? And is the plane still "at rest" at these values?
The static thrust value is with the engine fixed in place in still air. Imagine the engine on an outdoor test stand, or on an airplane with the brakes on or tied down. As soon as the airplane starts to move, the thrust value declines and the static thrust value no longer applies. The static thrust value at sea level is used because it provides a consistent baseline reference, because once the engine starts to move, additional variables come into play.
The lower chart is showing the relative decline in thrust with speed for the range of engines between pure jet (bypass ratio = 0) to very high bypass (5-6), to illustrate the impact of bypass ratio on the decline in thrust with speed. "Static thrust" is implied at the speed value of 0 Mach (airspeed is zero, so you are "Static" by definition) as a value of 1 and declines in thrust from static are a value less than 1. So you can interpret the left edge of the chart as "static thrust" and everything to the right as (moving) thrust, with the curves going down and recovering (or not) with increasing Mach #.
It's just illustrating relative effects, so with 6 engines that have different bypass ratios, their decline with speed, from the engine's Sea Level Static Thrust (whatever it is, represented by the value of 1), is shown by each curve. Following the curves as an example, the pure jet has only lost about 7% of its rated static thrust with speed at Mach 0.5, whereas the very high bypass fan has lost over 40% of its static thrust value at Mach 0.5, in the same atmospheric conditions.
Turbofans use a gas turbine core to drive the low pressure fan which produces most of the thrust by moving large volumes of air. This requires large amounts of torque which the low pressure turbine extracts from the hot exhaust gases exiting the gas turbine's core.
To develop more power you burn more fuel which means more heat. Low pressure gas turbine blades are usually rated over 2000F meaning they will melt if exposed to too much heat.
The reason for the rated 5 minutes of thrust is simply because the turbine blades will overheat causing premature wear and possibly engine failure. To produce the rated thrust the turbofan is at sea level in stationary air hence the term "static".
As the forward airspeed increases and the density of the air reduces due to increased altitude the rated thrust will go down which is not represented in the chart.