I've been reading that Phase Noise dominates in frequencies close to the carrier, while Amplitude Noise is more dominant in higher offsets. Could someone explain this?
This is kind of misleading. The blue line is the noise floor. Signals below the noise floor cannot be seen as peaks on the plot.
The carrier (if there is one) has a peak, and it falls off as you move away from the peak. A perfect sinusoid would be a straight vertical line. But in reality, there is variation in the frequency, and that causes the carrier to have a spread out shape.
So what you are calling phase noise is just part of the signal that is not at the center frequency. Basically, the statement that "phase noise dominates close to carrier" just means that the sloping sides of the peak are above the noise floor.
Phase noise is related to jitter or spectral purity. It isn't noise in the same sense as the noise floor.
Brownian or white noise is equal power per unit bandwidth which depends on spectal BW and becomes a variable noise floor of the spectrum analyser since the user controls the IF and video bandwidth.
Normally an oscillator suppress AM due to amplitude feedback unless there is coherent amplitude modulation and narrow band filtering. Since PM is the modulation of phase of the oscillator about centre, it is often shown with carrier rejected at the far left and upper side PM reducing to the right,
The dotted line is simply the actual curve between two asymptotic slopes.
Amplitude noise is the harmonics of the oscillator whether it is sine or square, so naturally far away off screen to the right.