Soft question: why define smooth manifolds intrinsically? A possibly naive question, but one I've been grappling with since starting Riemannian geometry. I have done some searching and I cannot find this question asked, yet it feels like a very natural one so I do apologise if I've missed it.
Why do we introduce the notion of a smooth manifold as intrinsic (locally Euclidean) when we can investigate Riemannian manifolds extrinsically as embedded in some higher dimensional Euclidean space (by virtue of the embedding theorems)? 
My reason for asking this question is not because I find this intrinsic approach unsatisfying, on the contrary my reason for asking is rather because I want to convey my fascination for its beauty to my friends and yet I very much struggle to give a cause-and-effect practical answer as to why we need, say, atlases and charts to tackle questions like curvature of a hyper-surface. 
I am wanting to say something along the lines of "well we cannot study the surface properly unless we exist on it as a kind of citizen of the surface without reference to anything else", but as of right now I cannot justify this statement at all (and maybe I shouldn't even be able to!).
 A: In the 19th century, three basic models of geometry were known to mathematicians:


*

*The Euclidean geometry which was more or less understood since the time of the Greeks.

*The spherical geometry which was developed by astronomers and mathematicians of the Islamic empire.

*The hyperbolic geometry which was a recent development and it was born out of a long-term struggle to understand the Euclid's 5th postulate better.


It turns out that all of these geometries, even when developed purely axiomatically, can be understood better once we study curved surfaces. For example, the Gauss-Bonnet theorem is one of these theorems that in its basic form connects the sum of the interior angles of a triangle on a curved surface with the curvature of the surface. Obviously, $2$-dimensional surfaces like sphere, torus, hyperboloids, etc. can be understood by us well because they can be embedded in $\mathbb{R}^3$ and we can use our tools developed for calculus in $\mathbb{R}^3$ to study them.
In 1818, Gauss was assigned to carry out a geodetic survey in the Kingdom of Hanover. During this time, he made measurements and invented tools like Heliotrope to do the job. It says that this job helped him formulate Theorema Egregium which he was very fond of it and was a starting point for developing calculus on surfaces.
Theorema Egregium (which means the Remarkable Theorem in Latin) says that an ideal ant (a two dimensional being) can understand the curvature of the surface it lives on by making local measurements. The significance of this is that this enables us humans to study higher dimensional surfaces by developing a calculus on the surface locally, instead of doing calculus on the ambient space. This also has physical implications. We humans are three dimensional beings, living in a universe which is probably four dimensional. Naturally, we are eager to understand the geometry of the universe around us. Theorema Egregium tells us that we are not completely hopeless and we can determine some local geometric properties of our universe just by local measurements.
For further reference, please read about Gauss–Codazzi equations, Riemannian geometry and General Relativity.
