Let $M$ be a smooth manifold, and $V,W$ smooth vector fields on $M$. If $\theta:\mathcal{D}\to M$ denotes the flow of $V$, where $\mathcal{D}$ is an open subset of $\mathbb{R}\times M$. The Lie Derivative of $W$ along $V$ at $p$ is given by the following
$$ (\mathcal{L}_V W)_p = \dfrac{d}{dt}\biggr|_{t=0} d(\theta_{-t}(p))(W_{\theta_{t}(p)}) = \lim_{t\to 0}\dfrac{d(\theta_{-t}(p))(W_{\theta_t(p)})-W_p}{t} $$
The tangent space at $p$ is a vector space, so $T_pM$ is endowed with a smooth structure. If we define $X(t) = d(\theta_{-t}(p))(W_{\theta_t(p)})$, this is a curve $X:(-\varepsilon,+\varepsilon)\to T_pM$ starting at $W_p$, since $X(0)=W_p$. The left hand side becomes
$$ (\mathcal{L}_V W)_p = \dfrac{d}{dt}\biggr\vert_{t=0}X(t) = X'(0)\in T_{W_p}(T_pM) $$
While the right hand side looks like some kind of Frechet derivative on $T_pM$. My question is, what is the highlighted part trying to say? This is from Lee's Introduction to Smooth Manifolds.
Highlighted part explains why can you switch order of derivative and $d\theta$. Next it argues that this is required result.
\begin{align} \frac d {ds}\bigg|_{s=0} d(\theta_{-t_0}) &\circ d(\theta_{-s})(W_{\theta_s(\theta_{t_0}(p))}) \\ &= \lim_{s\to 0} \frac{ d(\theta_{-t_0}) \circ d(\theta_{-s})(W_{\theta_s(\theta_{t_0}(p))})- d(\theta_{-t_0})(W_{\theta_{t_0}(p)})}{s} \\ &= \lim_{s\to 0} d(\theta_{-t_0}) \left(\frac{d(\theta_{-s})(W_{\theta_s(\theta_{t_0}(p))})- W_{\theta_{t_0}(p)}}{s}\right) \\ &= d(\theta_{-t_0}) \left(\lim_{s\to 0}\frac{d(\theta_{-s})(W_{\theta_s(\theta_{t_0}(p))})- W_{\theta_{t_0}(p)}}{s}\right) \\ &= d(\theta_{-t_0})( (\mathcal{L}_V W)_{\theta_{t_0}(p)}) \\ \end{align}
In second equality we used linearity of $d \theta_{-t_0}$. Next independence $d \theta_{-t_0}$ of $s$ and continuity of linear function on finite dimensional space (not emphasized by author) allows third equality. Final equality is definition of Lie derivative, where second vector field is $p \mapsto W_{\theta_{t_0}(p)}$. Since $\theta_{t_0}$ is locally a diffeomorphism, it is the same as $ (\mathcal{L}_V W)_{\theta_{t_0}(p)}$.