Aside from @mmesser314’s nice answer let me mention some other results. From the classification of surfaces, we know that the only (non-empty) 2-dimensional compact boundaryless simply connected smooth manifold is $S^2$. Said differently:

if $\Sigma$ is a non-empty 2-dimensional, compact, boundaryless, simply-connected topological (resp. smooth) manifold, then $\Sigma$ is homeomorphic (resp. diffeomorphic) to $S^2$.

So, the 2-dimensional sphere is the only thing you have. Note that this is a purely (differential) topological result. This makes no reference whatsoever to curvature or embeddings or whatsoever.

A standard fact about the sphere $S^2$ is that with its usual unit round metric tensor, it has constant sectional curvature $+1$ (this is a fully intrinsic notion). In fact, a nice topological result tells us something more: there is NO connection on $S^2$ with zero curvature… or said differently, $S^2$ doesn’t admit flat connections, or even more plainly, there is no sense in which $S^2$ is intrinsically flat. Once again, these are purely intrinsic statements.

Next, suppose we have an embedding of $S^2$ into a flat manifold $M$ (i.e $M$ is a manifold equipped with a flat connection). Then Gauss’ equation combined with the above result tells us that the extrinsic curvature (i.e the shape tensor) has to be non-zero. Roughly speaking, Gauss’ equation tells us how the ambient (intrinsic) curvature is related to both the intrinsic and extrinsic curvature of a submanifold, (i.e it relates three quantities). So if one of them is zero (e.g the ambient curvature) and one of the them is non-zero (the intrinsic curvature of the submanifold) then the other has to be non-zero as well (the extrinsic curvature of the submanifold).

But anyway even before all of these more ‘advanced’ results, I think you need to first of all appreciate the basic definitions of smooth manifold, (semi-)Riemannian metric, Levi-Civita connection, shape-tensor etc so that you understand what exactly the intrinsic notions are (and you’ll see that the extrinsic notions, even though historically the first ones discovered/introduced, are only introduced very late on in modern treatments of geometry).

Aside from @mmesser314’s nice answer let me mention some other results. From the classification of surfaces, we know that the only (non-empty) 2-dimensional compact boundaryless simply connected smooth manifold is $S^2$. Said differently:

if $\Sigma$ is a non-empty 2-dimensional, compact, boundaryless, simply-connected topological (resp. smooth) manifold, then $\Sigma$ is homeomorphic (resp. diffeomorphic) to $S^2$.

So, the 2-dimensional sphere is the only thing you have. Note that this is a purely (differential) topological result. This makes no reference whatsoever to curvature or embeddings or whatsoever.

A standard fact about the sphere $S^2$ is that with its usual unit round metric tensor, it has constant sectional curvature $+1$ (this is a fully intrinsic notion). In fact, a nice topological result (e.g this or this) tells us something more: there is NO connection on $S^2$ with zero curvature… or said differently, $S^2$ doesn’t admit flat connections, or even more plainly, there is no sense in which $S^2$ is intrinsically flat. Once again, these are purely intrinsic statements.

Next, suppose we have an embedding of $S^2$ into a flat manifold $M$ (i.e $M$ is a manifold equipped with a flat connection). Then Gauss’ equation combined with the above result tells us that the extrinsic curvature (i.e the shape tensor) has to be non-zero. Roughly speaking, Gauss’ equation tells us how the ambient (intrinsic) curvature is related to both the intrinsic and extrinsic curvature of a submanifold, (i.e it relates three quantities). So if one of them is zero (e.g the ambient curvature) and one of the them is non-zero (the intrinsic curvature of the submanifold) then the other has to be non-zero as well (the extrinsic curvature of the submanifold).

But anyway even before all of these more ‘advanced’ results, I think you need to first of all appreciate the basic definitions of smooth manifold, (semi-)Riemannian metric, Levi-Civita connection, shape-tensor etc so that you understand what exactly the intrinsic notions are (and you’ll see that the extrinsic notions, even though historically the first ones discovered/introduced, are only introduced very late on in modern treatments of geometry).