How does the watch orbiting Earth "know" it has to be the slower one? Pardon me for asking in layman's terms, but I find it the quickest.
When an astrounaut's watch has orbited earth in high speed for a sufficient amount of time, it is delayed by a detectable amount of time compared to an identical watch left back on Earth. Ok. Relativity.
But since the spacecraft, or earth are not tethered to a tangible point in space, we could perceive this scenario as Earth orbiting the spacecraft, and thus observe that the Earth-based watch is running slower. 
In other words, how do we decide, that it is the rocket moving fast around Earth, and not otherwise? And so, how does the astronaut's watch, and all satellites in orbit, for that matter, "know" it is their time, that should be slower?
 A: If you want to talk about special relativity you have to learn about it.
Inertial frames are important to special relativity:

An inertial frame of reference in classical physics and special relativity possesses the property that in this frame of reference a body with zero net force acting upon it does not accelerate; that is, such a body is at rest or moving at a constant speed in a straight line

...

All inertial frames are in a state of constant, rectilinear motion with respect to one another; an accelerometer moving with any of them would detect zero acceleration. Measurements in one inertial frame can be converted to measurements in another by a simple transformation (the Galilean transformation in Newtonian physics and the Lorentz transformation in special relativity

So the simple answer is that it is only inertial frames where time and space can be transformed to each other with no measurable effect.
Rotating frames, as the rocket going around the earth , are not inertial frames. This means that one can decide which frame is rotating and which is not. The earth frame is not rotating, the rocket is, thus the relativistic effects can display themselves making corrections to the rocket timing. This is taken into account in the GPS measurements:

Because an observer on the ground sees the satellites in motion relative to them, Special Relativity predicts that we should see their clocks ticking more slowly (see the Special Relativity lecture). Special Relativity predicts that the on-board atomic clocks on the satellites should fall behind clocks on the ground by about 7 microseconds per day because of the slower ticking rate due to the time dilation effect of their relative motion

goes on to discuss general relativity effects . Keep in mind that GPS works .
A: It is misconception that this has to do with the twin paradox. The twin paradox is a paradox because both of them could say the other is moving relative. Now this is because speed is symmetrically relative. Though, in your case, acceleration is absolute.
The clock on the spaceship and the clock on ground has time dilation because of:


*

*SR, the relative motion of the spaceship relative to a fixed point on the ground (the clock that is at rest on the ground). It is very important to understand that the relative motion of the spaceship is relative to the clock on the ground, and though the ground is moving too (Earth rotating), the spacecraft is moving relative to that fixed point on the ground where the ground clock is. The spacecraft is is orbital, which is, a circular motion, thus, acceleration. Acceleration is absolute and in this case the spaceship is accelerating relative to the fixed point on the ground (where the ground clock is).

*GR, the spaceship is in a area where the stress-energy is very little compared to the ground on Earth (where stress-energy is much stronger)
The SR effect would make the clock on the spaceship tick slower. The GR effect would make the clock on the spaceship (orbiting Earth) tick faster.
The two effect's net is that the GR effect is dominant, and the clock on the spaceship (orbiting Earth) ticks faster.


or clocks on GPS and Galileo satellites running slightly faster.


https://en.wikipedia.org/wiki/Time_dilation
