I don't know if the name of the theorem is absolutely correct but my book states it as (m,n) theorem. 
If $m.OA$ and $n.OB$ are two forces that are acting towards $OA$ and $OB$, and $C$ is a point on $AB$ such that $|AC| : |BC| = n : m$ , then the resultant force of the given forces is $(m+n).OC$.
Proof : Applying the triangle law of vectors for $\triangle OAC$ and $\triangle OBC$ we get $$OA + AC = OC....(1)$$$$OB + BC = OC....(2)$$multiplying (1) with m and (2) with n and adding them we get $$mOA + mAC + nOB + nBC = (m+n)OC$$we know that $|AC| : |BC| = n : m$ therefore, $m|AC| = n|BC|$ but from the picture it's clear that the direction of AC and BC are opposite. So they cancel out giving us,$$mOA + nOB = (m+n)OC$$
I get what the theorem is trying to say. But didn't get intuitive sense why what the theorem says actually works. I would like to have an intuitive explanation (an example might be helpful) about this theorem.
The theorem which you are referring to seems to be the Angle Bisector Theorem.
In the case of forces it doesn't apply since for parallelogram addition rule the resultant vector force passes throught the middle point between A and B.