I am not understanding how this has dimension $3$, but there are six components in each vector. If $3$ vectors span the space, why are there more than $3$ components in each vector? I thought for a set to be a basis there should be $n$ vectors and $n$ components in the vectors so that we may express the set as a square matrix.
The space does have 6 dimensions as you pointed out. The points that are solutions in that space, however, only make up a 3 dimensional subspace.
Using some smaller space examples we could look at the line
y=x in 2-dimensions and note that the solution space is only 1 dimension (the solution is a 1 dimensional shape (a line) in a 2 dimensional world).
Or if we look at a plane
z=x+y it'll be a 2-dimensional shape/solution in a 3-dimensional world.