TITLE: Understanding definition of differentiable manifold
QUESTION [5 upvotes]: I am currently studying a basic course on differentiable manifolds.I have read the following definition of differentiable atlas and manifolds:

Definition. Let $ \mathcal{A} = {(x_{\alpha},U_{\alpha})}_{\alpha \in A}$ be an atlas on a topological manifold $M$. Whenever the overlap $U_\alpha \cap U_\beta$ between two chart domains is nonempty we have the change of coordinates map $x_\beta \circ x_\alpha^{-1} : x_\alpha(U_\alpha \cap U_\beta) \to x_\beta(U_\alpha \cap U_\beta)$. If all such change of coordinates maps are $C^r$-diffeomorphisms, then $\mathcal {A}$ is called a $C^r$-atlas and a manifold endowed with maximal differentiable atlas is called differentiable manifold.

I really find it hard to understand the following:
Is there any intuitive idea from which the definition of diffferentiable atlas and manifolds emerge? How the local diffeomorphisms $x_\beta \circ x_\alpha^{-1} 
 : x_\alpha(U_\alpha \cap U_\beta) \to x_\beta(U_\alpha \cap U_\beta)$ allow us to define a differential structure on manifold globally?
Moreover,why is it required for atlas to be maximal in order to define differentiable manifold

REPLY [3 votes]: The user @M.Winter gives a very good explanation of why we have this maximal atlas. As far as the transition maps $x_{\beta} \circ x_{\alpha}^{-1}$, we require these to be diffeomorphisms because smoothness isn't invariant under homeomorphism. For instance, if $f: M \to  \mathbb{R}$ is smooth, we mean that given any $p \in M$, there exists a local chart $(\phi,U)$ such that $f \circ \phi^{-1}: U \to \mathbb{R}$ is smooth. 
Sometimes it is very useful to change your coordinates to make the problem simpler, but in order to do this without loss of generality, we need to know that if the problem says $f$ is smooth, then this is independent of the charts on $M$ that we use. This is what the transition map condition allows us to do. I hope this helps.