More a survey of related things than an answer, but here goes.
Let's write $D(n)$ for the space of compactly supported diffeomorphisms $\mathbb R^n\to \mathbb R^n$. A reasonable guess might be that this is contractible, but this is true only for very small values of $n$.
The space $Diff(S^n)$ of diffeomorphisms $S^n\to S^n$ contains the Lie group $O(n+1)$. A reasonable guess might be that the inclusion of $O(n+1)$ is a homotopy equivalence, but no. In fact, this is equivalent to the first guess: The subgroup of $Diff(S^n)$ consisting of diffeomorphisms supported in the complement of a given point may be identified with $D(n)$, and the multiplication map $D(n)\times O(n+1)\to Diff(S^n)$ (which is not a group homomorphism) is an equivalence. You can see this by comparing $O(n+1)$ with the space of cosets $Diff(S^n)/D(n)$. Thus $D(n)$ is what you might call the exotic part of the homotopy type of $Diff(S^n)$. It is also equivalent to the space of all diffeomorphisms $D^n\to D^n$ fixing the boundary pointwise.
Introduce one more player: the space of all compactly supported diffeomorphisms $\mathbb R^n\times \lbrack 0,\infty )\to \mathbb R^n\times \lbrack 0,\infty )$. Call this $P(n)$. It fibers over $D(n)$, and the fiber is equivalent to $D(n+1)$. It is equivalent to the space of "pseudoisotopies" of $D^n$.
The statement that every diffeomorphism of $S^n$ extends to $D^{n+1}$ means precisely that $P(n)\to D(n)$ induces a surjection on components. This is in principle weaker than the statement that $D(n)$ is connected, but it turns out that (at least for most values of $n$, maybe all?) $P(n)$ is connected.
In low dimensions the story is this:
$D(0)$ is a point.
$P(0)=D(1)$ is contractible because it is convex: convex linear combinations of order-preserving
diffeomorphisms of the line are again order-preserving diffeomorphisms.
$P(1)\sim D(2)$ is contractible. I am aware of two approaches to this:
(1) I believe that when Smale (re-)proved this he used the Poincare-Bendixson Theorem. The crux is that, given a compactly supported field of tangent lines in the half-plane $y\ge 0$ transverse to the line $y=0$, if you follow it from $y=0$ you will get all the way up and not get trapped in some spiral.
(2) Complex analysis. I always imagine that the following works, but I'm not sure of the details: The space of Riemannian metrics on $S^2$ is contractible because it is convex. The space of conformal structures on $S^2$ is contractible because its product with the space of positive functions on $S^2$ is that space of metrics. The group $Diff(S^2)$ acts on this contractible space. It acts transitively, by the uniformization theorem, and presumably in a strong enough sense that this implies that the subgroup preserving the standard conformal structure is equivalent to the whole. This subgroup (Moebius transformations plus complex conjugation) is equivalent to its maximal compact subgroup $O(3)$.
Thus $P(2)\sim D(3)$. Smale conjectured that this is contractible. Hatcher proved it.
Thus $P(3)\sim D(4)$. I don't actually know anything about this space.
For large values of $n$ (I forget how large):
The space $P(n)$ is connected, by a theorem of Cerf. But this does not mean at all that $D(n)$ is connected: we have an exact sequence $\dots \to\pi_1D(n)\to \pi_0D(n+1)\to \pi_0P(n)\to \pi_0D(n)$.
$\pi_0D(n)$ is the Kervaire-Milnor group of homotopy $(n+1)$-spheres, known to be finite and frequently nontrivial: You can make a smooth manifold homeomorphic to $S^{n+1}$ from any element of $\pi_0D(n)$ by gluing two hemispheres together. By the $h$-cobordism theorem, you get all homotopy spheres in this way; and it's elementary to see that two elements give the same thing if and only if they differ by something in the image of $\pi_0P(n)$.
There is an important map $P(n)\to P(n+1)$. Hatcher showed, and Igusa proved, that it is about $\frac{n}{3}$-connected. In the stable range $P(n)$ is essentially the Waldhausen $K$-theory of a point, which is rationally the same as the algebraic $K$-theory of $\mathbb Z$, and this implies plenty of elements of infinite order in the homotopy groups of $P(n)$ and $D(n)$ in degrees less than about $\frac{n}{3}$.
Best Answer
R.Palais, Morse theory on Hilbert manifolds (main Theorem of ยง12). As you will see, in the infinite dimensional setting the construction looses nothing in clearness.