Why is a circle in a plane surrounded by 6 other circles?
The short answer is "because they don't work," but that's kind of a copout. This is actually quite a deep question. What you're referring to is sphere packing in two dimensions, specifically the kissing number, and sphere packing is actually quite a sophisticated and active field of mathematical research (in arbitrary dimensions).
Here's one answer, which isn't complete but which tells you why $6$ is a meaningful number in two dimensions. The packing you refer to is a special type of packing called a lattice packing, which means it comes from an arrangement of regularly spaced points; in this case, the hexagonal lattice. The number $6$ appears here because the hexagonal lattice has $6$-fold symmetry. So a natural question might be whether one can find lattices in two dimensions with, say $7$-fold or $8$-fold symmetry, since these might correspond to circle packings with more circles around a given circle. (Intuitively, we expect more symmetric lattices to give rise to denser packings and to packings where each circle has more neighbors.)
The answer is no: $6$-fold symmetry is the best you can do! This is a consequence of the crystallographic restriction theorem. The generalization of the theorem to $n$ dimensions says this: it is possible for a lattice to have $d$-fold symmetry only if $\phi(d) \le n$, where $\phi$ is Euler's totient function.
The generalization implies that you still cannot do better than $6$-fold symmetry in $3$ dimensions. There are two natural lattice packings in $3$ dimensions, which both occur in molecules and crystals in nature and which both have $6$-fold symmetry, and it turns out that these are the densest sphere packings in $3$ dimensions. It also turns out that they give the correct kissing number in $3$ dimensions, which is $12$ (see the wiki article).
In $4$ dimensions, the kissing number is $24$, and I believe the corresponding packing is a lattice packing coming from a lattice with $8$-fold symmetry, which is possible in $4$ dimensions. In higher dimensions, only two other kissing numbers are known: $8$ dimensions, where the $E_8$ lattice gives kissing number $240$, and $24$ dimensions, where the Leech lattice gives kissing number $196560$! These lattices are really mysterious objects and are related to a host of other mysterious objects in mathematics.
A great reference for this stuff, although it is a little dense, is Conway and Sloane's Sphere Packing, Lattices, and Groups (Wayback Machine). Edit: And for a very accessible and engaging introduction to symmetry in the plane and in general, I highly recommend Conway, Burgiel, and Goodman-Strauss's The Symmetries of Things.
The question is equivalent to asking why 6 equilateral triangles fit together exactly around a point, with no additional room left over. The answer is "because the Euclidean plane is flat", a condition equivalent to triangles having angle sum of 180 degrees (half the angle around a point), so that each vertex of a symmetrical triangle has 1/3 of half of a full rotation = 1/6. That the six-circle arrangement exists for any radius is also a special feature of Euclidean geometry: scale invariance.
For flat surfaces such as a cylinder (rolled up plane) or flat torus (as in the Asteroids video game) the perfect 6-circle configurations exist only for small enough radius of the circles. These geometries are, in small regions, the same as the Euclidean plane but differ "globally", e.g. there is a maximum distance between points of the torus. So the magic number of 6 is really about local flatness (absence of curvature) and having this configuration for circles of all size is a global question about the space in which the circles live.
In hyperbolic geometry there are tesselations of the plane by equilateral triangles with angle $180/n$ at each vertex, for any $n \geq 7$. In the picture of the $n=7$ triangular tesselation at http://www.plunk.org/~hatch/HyperbolicTesselations/3_7_trunc0_512x512.gif (triangles in white with the dual tesselation by heptagons shown in blue) if at each triangle vertex you draw a circle inscribed in its heptagon, there will be 7 circles exactly surrounding each circle, with all circles of the same radius. The same type of configuration exists for any $n$ and suitable radius of the circles. So the flatness condition is necessary; the theorem is false in negatively curved two-dimensional geometry. Six is also not the correct number for spherical geomety, and both spherical and hyperbolic geometry lack a radius-independent "number of circles that fit around one circle".
In the geometry of surfaces, having 5-or-fewer as the local number of circles that can be fit around a single circle characterizes positive curvature, as in spherical geometry. Having more-than-6 fit, or extra room when surrounding one circle by six, is a characterization of negative curvature, as in hyperbolic geometry. This is a statement about the local geometry of general 2-dimensional surfaces, and does not assume the surface has the same amount of curvature at all points, as would be the case for the spherical and hyperbolic analogues of Euclidean plane geometry, where there is a homogeneity assumption that "geometry is the same at all points". Having exactly 6 circles fit perfectly is a characterization of locally Euclidean (that is, zero curvature) geometry. If you don't know what curvature is in a technical sense, for purposes of this discussion it is (for surfaces) a number that can be associated to any point on the surface, and curvature being zero in a region of surface is equivalent to the ability to make a distance-preserving planar map of that region. Impossibility of doing this for surfaces of nonzero curvature, such as the sphere which is positively curved, was Gauss' Theorema Egregium which ruled out perfect flat cartography of the Earth.
Flatness plays the same role in the higher dimensional, sphere-packing interpretation of the question that Qiaochu suggested. The necessary packings exist only in a limited set of dimensions. The reason is that exactly surrounding a sphere of radius $r$ by $k$ spheres of that radius means more than a kissing number of $k$, the optimal kissing configuration should also be rigid (tight enough that the spheres cannot be moved). Rigidity is false in three dimensions and is believed to be false in most other dimensions but in those cases where it is known or suspected to be true (i.e., dimensions $d = 2, 8, 24$ and possibly a few others) the existence of a configuration with the same set of equalities between various interpoint distances as in the optimal kissing arrangement (in flat Euclidean space $\mathbb{R}^d$)should force a homogeneous $d$-dimensional geometry to be flat. This is because deforming the curvature of the space in the positive direction would reduce, and negative curvature would increase, the freedom to position such spheres around a central sphere.
The centres of three circles of radius r just touching one another will form an equilateral triangle of side length 2r. Exactly 6 such non-overlapping equilateral triangles will be formed in this way as you keep adding more circles around the edge of the central circle. Why 6? These equilateral triangles will have a common vertex at the centre, each forming a 60 degree angle there. And as, you pointed out, "there are 6*60=360 degrees in a circle."