Suppose that $M$ is a smooth manifold with an atlas $A$, and $p$ is a point of $M$. pick a smooth chart $(U,\phi)$ such that $p \in U$, then a trick we often use is that we can always let $\phi(p) = 0$. The reason is that if $\phi(p) \neq 0$, we can simply compose it with a translation map.
My question is, if we do compose it with a translation map, how am I able to guarantee that this composition is still in the original atlas $A$? If it is not in $A$, then how am I able to conclude that this change of charts will not have any impact on the structure of $M$?
Handwavey: Translation is invertible.
More formally, you may replace your atlas $A$ with the extended atlas $A \cup \{\text{this translated version of $(U,\phi)$}\}$. It is easy to show that this translated version of $(U,\phi)$ is compatible with $(U,\phi)$ (because translation is invertible) and then that it is compatible with the rest of the atlas. This means appending this chart does not change the manifold.
Using more machinery, you could argue that each maximal atlas containing your original atlas contains the translated version of $(U,\phi)$. (Once again, the invertibility of translation is important.)