Take a step back and look at how loops are concatenated. Most of the magic takes place in the domain. A circle, which acts as domain for the product loop, is stretched to twice its length, wrapped around the first circle, then around the second. Compose this with the two loops in question, and the result is the third loop, thus implementing the group operation known as π1. We would like to define a similar operation for spheres.
The following assumes 2 dimensional spheres, i.e. π2, but this generalizes to n dimensions. It's just easier to picture 2 spheres in 3 space.
Let f and g be continuous functions from the sphere into a space Y. These are pointed spaces, hence f and g map the base point of the sphere to e, the base point of Y. Remember that f and g represent pointed homotopy classes of S2 into Y.
The idea is intuitive. Take a sphere and squeeze its equator in to the center, giving two tangent spheres. Call this function μ, a continuous map from one sphere onto two tangent spheres. Let f act on the lower sphere and let g act on the upper sphere. Compose this with μ to get h, a map from the sphere into Y.
That's the idea, but if we want to equate it with the unraveling process, we need to be a bit more precise. Let's start by describing μ.
think of S2 as the suspension of S1, where S1 is the equator. Remember, the suspension raises the circle up to the wall of a cylinder, closes the top and bottom to make a sphere, and compresses the base arc (derived from the base point of the circle) down to a single point, which becomes the base point of the suspension. Compressing the base arc to a base point stretches and shrinks the sphere, but the result is still a sphere.
Start with two disjoint spheres, Sf and Sg. We will (eventually) let f act on Sf, and g act on Sg. Make the two equators coplanar, and sew the two spheres together at their base points. thus t, the point of tangency, is now the base point for Sf and Sg.
Here's something that you don't want to miss. The two tangent spheres are the suspension of the two tangent circles, i.e. the two equators. Raise these circles up to tangent cylinders that share a common line segment, then close up the top and bottom, and finally, compress the common base arc down to t, the common base point.
If h, a continuous function from S2 into Y, is the "concatenation" of f and g, let Sh act as its domain. This too is a suspension of its equator. Let b be its base point, at 0 latitude and 0 longitude.
Compress the great circle of Sh running through 0 and 180 degrees longitude. This crunches the sphere into two smaller spheres, tangent at the point b. Our function μ is this quotient map, followed by the natural map onto Sf and Sg, with b mapping to t. Compose this with f and g to get h.
Now for the magic. Unravel f g and h, so they become loops in ω(Y). The equator of Sh wraps around the equator of Sf, then the equator of Sg. This is the mechanism that concatenates two loops together. In other words, the unraveling transformation respects concatenation. Composing f and g to produce h, int the manner described above, can be accomplished by unraveling, concatenating, and reraveling. The two worlds run in parallel. This isn't a rigorous proof, but I think it's pretty intuitive, so I'll let you fill in the details.
This correspondence holds at every level, hence πn corresponds to π1. Concatenation, as implemented by squeezing a great circle down to a point and mapping one sphere onto two, produces a group operator that coincides with the fundamental group of the n-1 order loop space of Y. As mentioned earlier, this group is abelian for n > 1, a fact that is far from obvious if you are given only the concatenation operator without its loop space equivalent.
Note that in one dimension, sphere concatenation is the same as loop concatenation. The "equator" of a circle consists of two opposite points. Pull them together and the circle becomes two tangent circles, ready to support two successive loops.