The gluon description is fine, but gluons are quanta of a quantum field theory, and you can view a field theory as a theory of fields or as a theory of particles. The particles description is noncausal, the particles go back in time, so that the field description is more intuitive for me, although for some reason many people say particles are more intuitive. I think that's because they think of these exchanged particles as classical particles going forward in time, which they are not.
The field picture of gluon exchange is that the strong force is exchanged by eight different copies of electric and magnetic fields, with complicated cross-charges, so that each field generates the other. This field is the glue field, and it is the main ingredient in QCD. The cross charges make all the difference, they cause the force to be short range.
The vacuum in electricity in magnetism is near zero field at long distances, with free photons propagating a long distance. For QCD this theory has the property that the field becomes random at a scale comparable to the proton radius. The randomization scale means that correlation functions fall off exponentially in the separation at this scale, and this leads to confinement.
The picture of confinement in the field point of view is the Wilson loop correlator of the Lattice gauge field. This means that you make a small grid, and define the gluon field in terms of a matrix to go from one point to another. You define a probability distribution for the matrices which, in the limit of small lattices and probability distributions concentrated near the identity matrix, reproduces a consistent continuous field, in that every path gets a matrix with a probability distribution which converges to a fixed distribtuion in the small lattice limit.
In this limit, you find that the matrices are completely random for large loops, and the cross over to complete randomness is the proton radius, give or take. This is explicitly done today on supercomputers, and it explains why the field is short range. The range of a field is the distance over which the statistical description forgets about a local change. If you change the field inside a proton, the field forgets the change outside the proton, and this means that the range is small.
Particle point of view
The field picture is somewhat better understood than the particle picture of confinement, although neither is at a point that they are persuasive to mathematicians. In the particle picture, due to t'Hooft, the screening of nuclear forces is due to something similar to a superconductor filling space. This is called the "dual superconducting model of quark confinement".
The picture is that when you have a magnetic monopole (isolated magnetic pole) inside a superconductor, the magnetic flux must go out, by magnetic Gauss's law, but the superconductor doesn't want the field around, because superconductors hate magnetic fields. So it squashes the magnetic field to a narrow tube, called a vortex, and this vortex begins and ends on the particle.
The idea is that the vacuum is filled with glue, and the glue is superconducting in a magnetic way, so that the electric charges end up with flux tubes in the same way magnetic charges do in a superconductor.
The dual superconducting picture is widely believed today, because it can be seen to work in some mathematical models with supersymmetry, but it is not understood very well in ordinary QCD. It is complementary to the field randomization picture, it gives the particle point of view regarding this. Notice that the particle view requires a coherent condensation in the vaccuum, a superconducter-like fluid that fills all of space.
The superconducting vacuum is different than the Higgs mechanism, because it is magnetic, not electric.