Intuitively this actually is very plausible, though you have to work your intuition a bit; fix a point in configuration space, this can be seen as a single point in something that locally looks like $\Bbb R^{2n}$ or $\Bbb R^{3n}$, but it is more convenient to think of $n$ distinct points of (something that locally looks like) $\Bbb R^2$ or $\Bbb R^3$.
A path through configuration space is a collection of paths based at these points. In the case of indistinguishable particles, we may also have paths starting at one point and ending at another, as long as at each point a path starts and a path ends. The only restriction is that the paths don't intersect for any given $t$, if $t$ is the (common) parameter of the paths. If the parameter runs from 0 to 1, the paths form a subset of $\Bbb R^2\times [0,1]$ and $\Bbb R^3\times [0,1]$. Two paths (in configuration space) are homotopic if there is a homotopy between all individual paths in $\Bbb R^2$ or $\Bbb R^3$ that leaves all end points fixed. In $\Bbb R^2$ it is quite easy to see that this gives the braid group.
Now for $\Bbb R^3$: you may be aware that in $\Bbb R^4$ there are no knots: all knots are homotopic to the circle. This may seem counterintuitive because we cannot visualize it, but in reality it's utterly trivial: it is the equivalent of having a circle in the plane with a point inside, it is not possible to continuously move the point outside the circle without intersecting it. However, when you add a dimension it is clear that this is not the case. Likewise for a circle in the $x-y$-plane around the $z$-axis is $\Bbb R^4$. This can be continuously untangled. Namely, move you circle in the $x-y$-plane into the parallel plane where $z = 0$, $w = 1$, if $w$ is the fourth coordinate. Every point of the circle has $w$-coordinate equal to 1. Move it within the $w = 1$, $z = 0$ plane to some large $x$. At every moment the $w$-coordinate of every point of the circle was 1, so it never intersected the $z$-axis.
Anyway, in 3D the braid of paths lives in $\Bbb R^4$ and can be untangled. Two paths in configuration space (i.e. braids of paths in $\Bbb R^3 \times [0,1]$) are homotopic to each other exactly when they connect the same initial and final points, i.e. when they define the same permutation. Paths twisting around eachother can all be transformed continously into each other.
