Every smooth manifold is assumed to be Hausdorff and second-countable.
Suppose $G$ is a Lie group, and $N,H\subset G$ are Lie subgroups such that $N$ is normal, $N\cap H=\{e\}$, and $NH=G$, where $NH=\{ab:a\in N, b\in H\}$.
How to show $\theta: H\times N\to N$ defined by $\theta(h,n)=hnh^{-1}$ is smooth?
By definition of Lie group, $G$ is a smooth manifold such that it’s differentiable structure is compatible with the operation of groups, so for each $g\in G$ the map
$\phi_g: G\to G$ that maps $h$ to $h \cdot g$ is smooth, $\psi$ that maps $h$ to $h^{-1}$ is smooth and the operation of group $\cdot : G\times G\to G$ is also a smooth map.
We consider the map $ (\cdot|_{H\times N}, \psi|_{H}) :H\times N\to G\times G$
that map each $(h,n)$ to $(hn, h^{-1})$.
This map is clearly smooth and your map is exactly
$\theta=\cdot\circ (\cdot|_{H\times N}, \psi|_{H})$
that is a composition of smooth maps, so it’s a smooth map.