I did an experiment to see if the angle of sliding is affected by mass. I had plates of similar weight and surface area and I increased their number successively. The data showed that there was no change in the angle, but I cannot understand why is that, is there a detailed scientific explanation? and why isn't the friction coefficient affected by mass?
The maximum amount of static friction which can be provided (the static limit) does depend on the mass. It is proportional to mass - or rather to the normal reaction between the surfaces in contact : $F=\mu N$. This is an empirical fact : it is what is usually observed to happen in the world, but there are exceptions. This is similar to Hooke's Law and Ohm's Law. The proportionality is an observational fact which often is a good approximation only over a range of values.
The constant of proportionality between the maximum friction which can be provided and the normal force is the coefficient of friction $\mu$. If friction did increase as normal force increased the law would be perhaps $F=\mu N+\nu N^2$, where $\nu$ is a second-order coefficient of friction.
Sliding starts when the component of weight down the plane equals the maximum force of static friction. As the angle of inclination is increased the component of weight down the plane increases, the normal force decreases and so the maximum friction force decreases. Increasing the mass increases these forces in the same proportion, so that the angle of sliding $\theta$ does not depend on mass :
There are various theories which attempt to explain the Laws of Friction, for example :
So friction=(mu)R =(mu)Wcos(a)
Body will start sliding when Wsin(a) is slightly grate than friction.
So angle a depends on coefficient of friction.
In comments you have said that coefficient of friction depends on mass. No it doesn't it depends only on the surfaces of two materials. As friction increases with mass you are probably assuming that coefficient increases. But as friction is product of coefficient and normal it increases with increased mass as normal depends on mass.
Static Friction is directly proportional to normal and the proportionality constant is the coefficient of friction.