The real motivation for the Lie derivative is doing differential calculus with vector fields. If we want fo differentiate the vector field $W$ in the direction of the vector field $V$, we take the flow of $V$ through time, use it to pull back $W$, and take the derivative at $t=0$.
To explain better, let $V$ and $W$ be vector fields on a smooth manifold $M$, with $\theta:M\times\mathbb R\supseteq\mathcal D\to M$ the flow of $V$. Because $\theta_t$ is a diffeomorphism from the set $\{p\in M| (t,p)\in \mathcal D\} := M_t$ to the set $M_{-t}$, with inverse $\theta_{-t}$, we can use it to pull back vector fields, as follows: define $(\theta_t^*W)_p$ by $$\theta{_{-t}*} W_{\theta_t(p)}\in T_pM.$$ We can then take the following limit$$\lim_{t\to 0}\frac{(\theta_t^*W)_p -W_p}t := \mathcal (L_VW)_p = \frac d{dt}\big |_{t= 0}(\theta_t^*W)_p$$ which we call the Lie derivative. It allows us to transport a vector field at one point to a vector at another, and find the rate of change of that transformation.
Why do we do this? Because it is a step towards generalizing analysis to an arbitrary smooth manifold. It allows us to take the derivative of a vector field "along" the flow of another.
However, there is a problem. Lie derivatives do not allow us to take directional derivatives along curves, as it leads to problems. This is why we use the notion of a connection.
(My source for the last comment is the following Math.SE question: Link. My main source generally is John Lee's Introduction to Smooth Manifolds.)
