why simple harmonic motion is projected as or compared with uniform circular motion ?
Because uniform circular motion is essentially "two dimensional simple harmonic motion."
Consider the shadow of an object moving in a circle when the light is shined "edge on".
(This essentially "collapses" or "ignores" one of the dimensions of the motion)
Well if one considers a cartesian coordinate system with origin at the center of the circle of the uniform circular motion, then can prove fairly easily that the x and y coordinates (of the particle that does circular motion), are doing simple harmonic motion. That is it will be x=R*cos(wt) and y=R*sin(wt) where w the angular velocity, R the radius of the circle.
So you can view uniform circular motion as composition of two simple harmonic motions of the same frequency, with phase differnce pi/2, and (thats important) of direction perpendicular to each other.
The usual form the solution is given in, that is in terms of a shifted cosine, can be interpreted as the x component of a body in uniform circular motion.
Separate names with a comma.