Things are not difficult if you put yourself in the right perspective.
"Curvature" is a mathematical concept. Like many mathematical concepts the word may sound like something a taxi driver in London or Jakarta may have heard, but it is really just a "formula". Likewise: "Work" may evoke ideas of money, strikes, unions, customers..., but in physics it is just a "formula".
So forget what you may think you know about "taxi driver curvature" and concentrate on the mathematical concept.
"Curvature" was introduced in all generality by Riemann in the late 19th century. It is basically a measure of the acceleration of neighboring geodetics (geodetics are the straightest possible curves that you can draw on a manifold). In GR the manifold is spacetime and geodetics are the trajectories of freely moving particles. So imagine yourself being transported in the space between 2 galazies. This is a very remote location, the closest star is a million light-years away. You are bored , so you throw 2 marbles in different directions. Then you measure the increasing distances between the 2 marbles. This distance is increasing, but at a constant rate. That is they are not accelerating relative to each other. You repeat the experiment with other marbles in other directions. No acceleration. So you say: there is no acceleration between any 2 neighboring geodetics, so there is no curvature, spacetime is flat in this area. Then you are brought somewhere else. Again, out of boredom you start exploring this new area of spacetime, you throw 2 marbles in exactly opposite directions and start measuring their separations. At first their separation increases, though not linearly, then it start decreasing until the 2 marbles collide into each other further away. This accelerating/decelerating behavior clearly indicates curvature and, by repeating the experiment in different directions, you can measure precisely the curvature present in this area of spacetime. It turns out that you are orbiting a planet and the first 2 marbles you threw in opposite directions were on the same orbit and met halfway around the planet. As you see there is no reason to talk about an "external space with n dimensions". All calculations are intrinsic to our 4D universe.
Gauss was the first to realize that you can investigate the shape of a 2D surface simply by making measurements on the surface itself. Riemann went on to generalize this idea to n-D surfaces (manifolds).
By the way, similar calculations are performed routinely if you want to measure the earth curvature (shape). Here you draw geodesics in space (straight lines, using lasers) between different locations and , via triangulations, you calculate the shape of the ground surface (hills, valleys). This is the science of geodesy.
"Intrinsic curvature" is the only one that counts in GR and is the most important in mathematics. You just call it "curvature". It is called intrinsic because it is calculated (and therefore "observed") by simply making measures "inside" the manifold. There is also the concept of "extrinsic curvature" in mathematics. It measures the way a submanifold is immersed in a higher dimensional manifold. The only way to measure this extrinsic curvature is to stand out of the submanifold and make measurements involving quantities which exist outside of the submanifold (normal vectors, for example). Since we have no way to stand out of our universe, this kind of curvature plays no role in cosmology (and physics in general).
Some geometrical examples:
take a 2D plane immersed in the usual 3D space. Draw diverging geodetics (straight lines) on the plane and calculate the intrinsic curvature. It is zero. Now look up the formula for extrisic curvature. It involves the variation of the normal vector to the plane. Since the normal does not change, the plane has zero extrinsic curvature too. Good, that sounds reasonable. Now gently bend the plane. Do not make corners or cuspids, just bend it. You did not stretch the plane, so distances on the plane did not change, thus the geodetics stayed the same. Conclusion: the bended plane has still zero intrinsic curvature. But now the normals change, expecially around the bend, so the extrinsic curvature is not zero anymore. That sounds reasonable: the bended plane has an extrinsic curvature. Did the little "ants" living on the plane notice that the plane was bent? No, since all distances on the plane stayed the same. Now go all the way and roll up the plane to make an infinitely long cylinder. Sure you have to cut part of the plane off and glue some parts together, but you do not stretch anything, so...the instrinsic curvature is still zero. Sounds weird? A cylinder has zero intrinsic curvature? Yes! Stop thinking like a taxi-driver. Just follow the formula. Do the little ants notice now that something has changed? You bet! The dishonest taxi-driver ants will notice it immediately and take the longest route from A to B, instead of the shortest :-) The physicist ants though will keep saying that the intrinsic curvature is still zero, but that the topology has now changed. Right, the topology, not the curvature. That's because , to make the cylinder, we had to cut and glue parts of the plane. However, they can discover this only by making "global", ie long range, measurements on the manifold (they have to go around the cylinder). The "local" measurements (geodetics) won't tell the difference. GR does not say anything about the topology of our universe. We are looking to see if we can find light from the same galaxy coming from opposite directions. So far nothing conclusive. We do see light from the same galaxy coming from 2 or more slightly different directions. This is due to gravitational lensing caused by another galaxy lying in the middle.