A "spline space" is a vector space. The elements of this vector space are the spline curves (or piecewise polynomials, if you prefer). So, like any finite-dimensional vector space, this space has a "dimension". That's what they're talking about.
The dimension of a vector space is the number of elements in any basis. Very roughly speaking, it's a measure of the "size" or "expressive power" of the space. In the case of a spline space, the dimension is mostly dependent on the number of entries in its knot sequence.
It's not related to the dimensionality (2D or 3D or whatever) of the control points.