One important thng to remember is that smooth manifolds were born as a means to talk about regular surfaces of euclidean space without needing to worry about an ambient space. There are several reasons why this is useful, for one physical motivation, consider the case of General Relativity. According to it, the universe is a 4-dimensional manifold which need not be euclidian, so physically, what would be an ambient space for the "space" itself? This is just one of the many reasons one would to want to work with an intrinsic definition of "regular surface".
Of course, you might wonder if indeed manifolds are really just regular in desguise. The answer is yes! In fact, we have embedding theorems that say that every manifold is diffemorpic to some regular surface of some Rn. This, at least to me, shows that the definition is good for what it intented to do: every regular surface is a manifold, but also, every manifold is a regular surface.
Of course, as with any other defintion, we can remove some requirements and end up with more general objects, which often times are also very interesting on their own. For instance, if you remove the smoothness of the change of coordinates, you end up with a topological manifold, which have been studied extensively.