The statement "$L_X Y=[X,Y]$ calculates changes of $Y$ along the integral curve of $X$" is not quite correct. Let me explain why.

Let $\phi_t^X$ denote the flow of $X$.

The key formula to understand the bracket (already mentioned above) is
$$[X, Y] = \left. \frac{d}{dt} \right|_{t=0} \left( (\phi_t^X)^* Y \right).$$
(Pay attention to the location of the star: $\psi^* X$ and $\psi_* X$ denote respectively the pullback and the pushforward of $X$ by $\psi$; one of them corresponds to a "passive" change of coordinates while the other corresponds to an "active" transformation. Confusing them would lead to a sign error.)

Here is what this formula says. Imagine you push the vector $Y$ along the flow of $X$ for some time $\Delta t$, and you compare it to the vector that is already sticking out of the point you have reached. You divide the difference by $\Delta t$, and you make $\Delta t$ tend to $0$; this gives you $[X, Y]$.

The crucial thing to understand here is that when you do this, **two** things happen:

Obviously as you move along the integral curve of $X$, the value of $Y$ changes. The rate of this change is one of the terms that comprise $[X, Y]$. This is what you must have thought about when you said that "$L_X Y=[X,Y]$ calculates changes of $Y$ along the integral curve of $X$". But this is only part of the story, because...

The flow of $X$ is **NOT** a translation, because $X$ need **NOT** be locally constant. (In fact, on a general manifold, neither the notion of "translation" nor that of a "locally constant vector field" make sense, because these notions *do* depend on the coordinate system you choose.) So the flow of $X$ can stretch, squeeze or rotate the manifold, and then it stretches, squeezes or rotates $Y$ correspondingly. This means that even if $Y$ is "locally constant" (in some coordinate system), the bracket can still be nonzero.

These two contributions account respectively for the two terms in the right-hand side of the formula
$$[X, Y] = \nabla_X Y - \nabla_Y X.$$
Check this! This is obvious for the first term, and requires some thinking for the second term.

If you do not know what the covariant derivative $\nabla$ is, you can think of vector fields on $\mathbb{R}^n$, and interpret $\nabla_X Y$ simply as "the directional derivative of $Y$ along $X$". This makes sense on $\mathbb{R}^n$, but not on an abstract manifold (if you try to define it with coordinates, you will get different values depending on what coordinate system you choose - unless you have an additional structure such as a Riemannian metric.)

The left-hand side, on the other hand, *always* makes sense, which is why it is introduced. The advantage is that it is invariant by diffeomorphisms (or, if you prefer, by change of coordinates). The drawback is that $(L_X Y)_x$ does not only depend on the value of $X$ at $x$, but on the value of $X$ on a whole neighborhood of $x$.