I cannot understand the concept of a divisor for an elliptic curve I am trying to understand ABE for a project of mine. I've grasped the concept of elliptic curves, but could not understand the meaning of divisors for functions on them. 

Could anyone help me with this? I mean, why would we want to define functions on an elliptic?  What is the benefit?
As @user94293 mentioned, divisors are a way to keep track of the zeroes and poles of a function. (Poles are related to projective coordinates).

Think of a function like $f(x) = x^2 - x + 6$. It has two zeroes: $-2$ and $3$. If I only gave you these two zeroes, you would be able to recreate the original function, up to a constant factor.

The reason divisors are used in pairing-based cryptography (and thus in ABE) is because the pairing function itself is defined as the function which has a particular divisor. This is enough since, as mentioned, if you have the divisor (and thus the zeroes), you can reconstruct the function.

In particular, the Tate pairing is defined as the function $\tau_r(P, Q) =  f_\tau(Q)^{(q^k-1)/r}$ where $f_\tau$ is a rational function (a function which is the ration of two functions) with divisor $r\langle P \rangle - r\langle \infty \rangle$, which means it has $r$ zeroes in point $P$ and $r$ poles in the point at infinity. This function is computed with Miller's algorithm.

In short, divisors are used to specify the pairing function used in pairing-based cryptography.