Intuitively, why does gain margin and phase margin infer instability in feedback system?
why increasing the "gain" of your system will cause instability


If you have a servo control mechanism and you set a demand, the servo should rotate (or move) to the position demanded and all is good. However, if you have too much gain, the servo rapidly moves off in the direction needed but overshoots the target due to the momentum built up in the rapid acceleration. This may eventually settle down or it may just continue to oscillate (unstable).


  why delaying a signal will cause instability? This doesn't seem to be
  very intuitive at all. Because if I delay my sine by 2π I get back my
  sine again.


You've basically described the reason why sustained oscillations might occur - basically you don't want a delay because a delay is likely to cause problems as you have described i.e. sustained oscillations.


  How does gain and delay translate into physical components inside of a
  feedback system (say a circuit). Intuitively, a gain is an op amp,
  what about a delay?


Gain might be from an op-amp or, you might have a totally digital control with ADCs and DACs. An op-amp might be used as an integrator rather than strictly speaking a gain stage AND, if you apply gain, integration and differentiation (maybe three op-amp circuits) you get a PID controller: -



Pretty picture taken from here (a public domain image).

If you study the above you will see it moves through three phases.


Purely applying gain until there is potentially too much overshoot but there is still a basic control inaccuracy
Then applying an integration term to improve the basic control accuracy but it risks creating too much overshoot
Applying a differential term to the above to restore sensible operation by dramatically reducing overshoot.



  Also, what does instability here refer to? Are we talking about the
  circuit going into oscillation or blow up behavior?


I think I've covered this.