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"Power" is the rate of change of energy. It's measured in watts, where 1 watt is 1 joule-per-second. If the aforementioned 2kg object is accelerated using 1 watt of power, it will take four seconds to reach the 4 joules where it's moving at 2 meters-per-second; if it's accelerated using 2 watts of power, it will only take two seconds to reach a speed of 2 meters-per-second.
But in the real world, there are also resistances to overcome, such as aerodynamic drag. Resistances apply power against the kinetic energy of an object. If a power source is trying to accelerate a moving object with 1 watt, but the object is also facing 1 watt from resistances, the object will not accelerate or decelerate, but rather continue moving along at its current speed.
Resistances generally become larger as speeds increase. If a power source is trying to accelerate a moving object with 2 watts, and it's only facing 1 watt of resistance, the object will accelerate until it's going fast enough that it's facing 2 watts of resistance. Similarly, if a power source is trying to accelerate a moving object with 1 watt, but it's facing 2 watts of resistance, the object will decelerate until it's only facing 1 watt of resistance.
On a bicycle, the rider's pedaling is a power source used to accelerate and maintain speed. Opposing the cyclist's motion are resistances like aerodynamic drag, rolling resistance, and gravity (if the cyclist is climbing).
And: reducing your resistances, such as by using fast tires that have less rolling resistance, allows you to maintain a given speed at lower power (and therefore lower effort).
When used carefully, a power meter can also be used to characterize how efficient your bike setup is. So if you do something like change your posture on your bike, you can check whether the change increased or decreased your aerodynamic drag.
Why does it matter that you can measure how hard you are training? Because it allows you to increase your training load incrementally so that you are improving your fitness steadily without overdoing it and getting injured. It also helps on race day to pace yourself at a steady effort regardless of course conditions (e.g. not going too hard into a headwind trying to chase a target average speed).
If you don't have power then a combination of HR and RPE are the next best thing, and even if you do have power then you should still pay attention to HR and RPE as they may well be telling you something that power isn't (e.g. even though you've comfortably held x watts a dozen times in training, today is a bad day for whatever reason - illness, fatigue, nutrition, etc - and you need to back off a bit)