So this entire discussion is in Goerss and Jardine's "Simplicial homotopy theory" and also in May's "Simplicial objects in algebraic topology". Also Curtis' papers and monographs are very nice and classical. One aesthetic reason that one may want simplicial homotopy groups is to show that we can calculate homotopy groups within the category of simplicial sets. Thus one sets up this machinery. I think that Milnor proved the comparison between simplicial and let's say topological (even though this is not quite accurate, I think something like CGHaus homotopy groups) groups.
Here is how you speak about homotopy groups in the context of simplicial sets: First you need the notion of a horn. A horn is the boundary of an n-simplex with a lost face. Now you must remember that simplicial sets have arrows on their edges, so we have a couple of different horns in each dimension. In dimension 2 for example, we have three different horns (and will need all three of these horns for our to define the fundamental group, in addition to a three dimensional horn to give associativity). So we now define a Kan complex as a simplicial set in which each horn may be filled out to any n-simplex that it is contained in (this is written as a lifting property).
So we will use this to define the fundamental groups, as the higher homotopy groups are analogous. Pick a basepoint, and two loops based at that point (if we do not want to talk about basepoints this discussion works for fundamental groupoids). This can be realized as a map of a two horn into the simplicial set. Now pick a horn filling (it doesn't matter which, they all differ by homotopy equivalence, namely an even higher horn fills this choice of two horns. This is similar to higher category theory.). The group operation of the two loops is the new loop created on the boundary of the two simplex. Considering the other two horn fillings will give left and right inverses in the group, which must be shown to be homotopy equivalent (by more horn fillings)
All the properties that you would expect of such composite can be shown to be true by more (generalized) horn fillings. These are called anodyne extensions. It turns out if you can fill all of your horns, you can fill all of your anodynes. This will show that composition is independent of choice of representatives.