Isn't the fact that a local paramterization has to be a homeomorhpism (locally) enough to preserve dimensions.
Around every point p on a submanifold you can find a local parametrization $\phi$ (with $\phi(q)=p$) with some conditions. However, what's the intuition behind the fact that you want the total derivative of the parametrization (evaluated in q) D$\phi(q)$ to be injective. How should I visualize this property? Why do you need this?
It is true that two open sets are homeomorphic if and only if they belong to Euclidean spaces with the same dimension.
When you introduce differentiability in your coordinate system you are looking for a smooth structure in your manifold and to apply the ideas of calculus on it. For example, study derivatives on it, define flows to explore the manifold, etc.
To do this you have to give a precise meaning to "flowing along the manifold". Thus, since a manifold is "twisted" you cannot compute derivatives along the usual lines, you have to go through directions tangential to the manifold. So, in few words, the condition on the rank of the derivative $D\phi (q)$, i.e. injectivity, is imposed to define the tangent space on the manifold at the point $\phi(q)$ and to guarantee that is well defined, i.e. independent of the coordinate system.
If we define $T_\phi(q) M:= D\phi(q)\left(\mathbb{R}^N\right)$ then it must be a $N$ dimensional space. But if $D\phi(p)$ is not injective then it deforms the domain into another linear space with lower dimention, implying that you cannot attach a tangent plane of the required dimension showing you that your manifold (e.g. surface) is not differentiable at the point.
In Munkre's book of Analysis on manifolds you can find further explanations of this and the proof that the injectivity in the derivative is fundamental to define properly the tangent space.
$\textbf{Note}$: From a technical point of view the infectivity in the derivatives of the coordinate systems is the equivalent version for embedded manifolds of the requirement of having differentiable transition functions in your atlas to define a general smooth manifold.