This is not the same question as What is the mathematical definition of a "figure six"? . Although it pertains to the same topic, and the same figure six. This is the statement of the theorem and the paragraph introducing the problem. Scroll to the bottom to get to the essential question. Taken from Edwards, Advanced Calculus of Several Variables. Chapter III.
Theorem 4.2 Let $M$ be a subset of $\mathbb{R}^{n}$. Suppose that, given $\mathbf{p}\in\mathit{M}$ , there exists an open set $U\subset\mathbb{R}^{k}(k<n)$ and a regular $\mathscr{C}^{1}$ mapping $\varphi:U\rightarrow\mathbb{R}^{n}$ such that $\mathbf{p}\in\varphi(U)$, with $\varphi(U^{\prime})$ being an open subset of $M$ for each open set $U^{\prime}\subset U$. Then $M$ is a smooth $k-$manifold.
The statement that $\varphi(U^{\prime})$ is an open subset of $M$ means that there exists an open set $W^{\prime}$ in $\mathbb{R}^{n}$ such that $W^{\prime}\cap M=\varphi(U^{\prime})$. The hypothesis that $\varphi(U^{\prime})$ is open in $M$, for every open subset $U^{\prime}$ of $U$, and not just for $U$ itself, is necessary if the conclusion that $M$ is a $k-$manifold is to follow. That this is true may be seen by considering a figure six in the plane\textemdash although it is not a $1-$manifold (why?); there obviously exists a one-to-one regular mapping $\varphi:(0,1)\rightarrow\mathbb{R}^{2}$ that traces out the figure six.
Edit to remove superfluous ramblings intended to show my effort at finding an answer.
The question is: Why is it required that every subset of the image of an open set of manifold coordinates be an open set of the manifold?
I believe I figured this out. By the definition of $\varphi[U^{\prime}]$ being an open subset of $M$ it follows that only points of $M$ in the image $\varphi[U^{\prime}]$ are allowed in the corresponding open subset $W^{\prime}\subset\mathbb{R}^2$.
In the image, the portion of the figure six colored green, $\varphi[U^{\prime}]$, has a corresponding open disk, $W^\prime$, satisfying the condition that it intersects $M$ only at points of $\varphi[U^{\prime}]$. The smaller portion of the curve, $\varphi[U^{\prime}_*]$, between the points touching the perimeter of the smaller, overlapping blue disk, $W^\prime_*$, intersects points of $M$ outside of $\varphi[U^{\prime}_*]$.