Let $d\in\mathbb N$, $k\in\{1,\ldots,d\}$, $M$ be a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$, $T$ be a $C^1$-diffeomorphism from $\mathbb R^d$ and $N:=T(M)$.
It's easy to see that $N$ is again a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$. In fact, if $\phi$ is a $C^1$-diffeomorphism from an open subset $\Omega$ of $M$ onto the open subset $U:=\phi(\Omega)$ of $\mathbb R^k$, then $$\psi:=\phi\circ\left.T^{-1}\right|_{\Omega'}$$ is a $C^1$-diffeomorphism from $\Omega':=T(\Omega)$ onto $U$.
However, how do we see that $\Omega'$ is an open subset of $N$?
Based on the discussion below this answer, this seems to be an easy application of the inverse function theorem. However, I don't get how it is applied exactly.
The inverse function theorem needs a function defined on an open subset of $\mathbb R^m$ to $\mathbb R^m$ for some $m\in\mathbb N$. The only function which came to my mind would be $\psi\circ T\circ\phi^{-1}$, but this function is simply the identity $\operatorname{id}_U$ on $U$. So, what do we need to do exactly?
I guess the desired claim follows from the following result:
Let $d_i\in\mathbb N$, $k_i\in\{1,\ldots,d_i\}$, $M_i$ be a $k_i$-dimensional embedded $C^1$-submanifold of $\mathbb R^{d_i}$ and $f:M_1\to M_2$ be $C^1$-differentiable at $x_1\in M_1$.
Assuming that $T_{x_1}(f):T_{x_1}\:M_1\to T_{f(x_1)}\:M_2$ is injective and $k:=k_1=k_2$ (if I'm not missing something, both assumptions together are equivalent to require that $T_{x_1}(f)$ is an isomorphism), we are able to apply the inverse function theorem and show that $\left.f\right|_{\tilde\Omega_1}$ is a $C^1$-diffeomorphism from $\tilde\Omega_1$ to $f(\tilde\Omega_1)$ for some $M_1$-open neighborhood $\tilde\Omega_1$ of $x_1$.
In fact, let $x_2:=f(x_1)$ and $\Omega_2$ be an $M_2$-open neighborhood of $x_2$. Since $f$ is continuous at $x_1$, $$f(\Omega_1)\subseteq\Omega_2\tag1$$ for some $M_1$-open neighborhood $\Omega_1$ of $x_1$. Since $M_i$ is a submanifold, there is a $C^1$-diffeomorphism $\phi_i$ from $\Omega_i$ to $U_i:=\phi(\Omega_i)$. Let $$\psi:=\phi_2\circ\phi_1^{-1}$$ and $u_i:=\phi_i(x_i)$. By assumption, ${\rm D}\psi(u_1)$ is injective (if I'm right with my remark above, it is even bijective) and hence, by the inverse function theorem, $\left.\psi\right|_{V_1}$ is a $C^1$-diffeomorphism from $V_1$ to $V_2$ for some $\mathbb R^k$-open neighborhood $V_1\subseteq U_1$ of $u_1$ and a $\mathbb R^k$-open neighborhood $V_2\subseteq U_2$ of $\psi(u_1)=\phi_2(f(x_1))$. Thus, $$\left.f\right|_{\tilde\Omega_1}=\phi_2^{-}\circ\left.\psi\right|_{V_1}\circ\left.\phi_1\right|_{\tilde\Omega_1}\tag2$$ is a $C^1$-diffeomorphism from $\tilde\Omega_1:=\phi_1^{-1}(V_1)$ onto $$\tilde\Omega_2:=\phi_2^{-1}(\underbrace{\psi(V_1)}_{=\:V_2})=f(\tilde\Omega_1)\tag3.$$ Since $\phi_i$ is continuous and $V_i$ is $U_i$-open, $\tilde\Omega_i$ is $\Omega_i$-open and hence a $M_i$-open neighborhood of $x_i$.