I want to understand the principal of least action intuitively, away from any mathematical proof.
Well, this question is rather opinion based but I will make an effort to answer it according to my intuition. The principle of least action is the fact that physical processes happening in nature must happen in the "least complicated" way possible.
What I mean by "least complicated" can be understood by an example: Suppose that a classical particle has to move inside an isotropic space from a point $A$ to a point $B$. Then according to the least action principle, it has to propagate on the straight line connecting these points. It must not follow any other way because it doesn't have to complicate things (since the space is isotropic for example).
In a similar way, all physical processes have to follow the least action principle just like the above example of the propagating particle in an isotropic space. This explanation is just my intuition on this subject. I am sure that you will get other beautiful explanations as well.