To apply a vector field $V$ to a function $g$ (at a point $p$), take the directional derivative of $g$ along $V$. This is to say flow $g$ along $V$ some small distance $\epsilon$, take the difference quotient, and let $\epsilon \to 0$.
The Lie bracket $[X,Y]$ is defined as the vector field given by $[X,Y]f = X(Yf) - Y(Xf)$. So, loosely speaking, we are infinitesimally flowing along $Y$, then $X$ and also along $X$, then $Y$, and taking the difference. Since subtracting is adding the opposite, we're flowing infinitesimally along $Y$, $X$, then $-X$, and finally $-Y$.
I'm assuming you want an informal description, not a formal reason to think that $[X,Y]$ is the flow along $X$, then $Y$, then back along $X$, then back along $Y$. If you want the latter, I'll have to rewrite this.