What is the origin and cause of the strong interaction which occurs between the nucleons? I have read that it is caused by the exchange of mesons but what then ultimately causes this meson exchange to occur? What is the real cause behind this strong force?
Both nucleons and mesons are composed of smaller particles known as quarks, which are elementary particles (meaning that, as far as we known nowadays, there are fundamental, instead of composed out of even smaller particles).
According to the Standard Model of Particle Physics, the quarks are subject to three sorts of interactions: electromagnetic interactions, weak interactions, and the strong interaction. The existence of the strong interaction among quarks is due to the existence of particles known as gluons, which provide these sorts of interactions. Why do quarks and gluons exist? Currently, no one knows (there certainly are proposals for this, but currently the Standard Model is the most fundamental we can go within Particle Physics).
As for the interaction between nucleons, it is understood as a residual effect of the interaction between quarks and gluons. In other words, one can roughly picture the nucleons to be trios of quarks bound together by the gluons, while mesons are pairs of quarks bound together by the gluons. As a consequence of the interactions between quarks and gluons, it follows that the nucleons and mesons will interact.
I should point out that, as far as I'm aware, the problem of understanding how precisely Quantum Chromodynamics (the piece of the Standard Model ruling the interactions among quarks and gluons) gives rise to nucleons, mesons, and other hadrons is still an open problem. In other words, while this explanation of mesons ans nucleons arising from quarks and gluons is widely believed to be correct, the mathematical details of how this process happens are still not understood.
For some extra information, you might be interested in reading a bit about color confinement.
Physics is a scientific discipline that has mathematical tools which answer how from certain principles and assumptions the mathematical formulation that fits the data, will also predict new phenomena. This is a progressive process. Like peeling an onion as accelerators increased in energy and particle physics developed, the quark model described particle data and the standard model was successful in fitting and predicting the majority of existing data in particle physics.
The meson exchange for describing the nuclear force is what we were taught during my graduate school years in the 60's :
In the 1950s and 60s more mesons were found in accelerator experiments and the meson theory of nuclear forces was extended to include many mesons. These models became known as one-boson-exchange models, which is a reference to the fact that the different mesons are exchanged singly in this model. The one-boson-exchange model is very successful in explaining essentially all properties of the nucleon-nucleon interaction at low energies.
That was long before the quark model. So the theoretical tools at the time were "very successful in explaining essentially all properties of the nucleon-nucleon interaction at low energies", and this of course is still true.
Why this was so at the time had the only answer :because the model fits mathematically the data , and is predictive.
With the accelerators going to higher and higher energies we have peeled the onion one more layer and discovered what the interactions are between quarks and have arrived at the strong colored force and quantum chromodynamics , and the standard model fits and predicts most data in particle physics.
The strength of the known forces is reflected in the coupling constants , and the colored force has small range but is very strong . This is what allows models that explain the emergence of mesons and even nucleons from a quark gluon plasma in high energy particle scatterings. The quantum chromodynamic mathematical tool of Lattice QCD can be used to show how the strong nuclear force emerges from the quark content of the neutrons and protons of a nucleus, which are full of quarks and gluons.
The spill over strong colored force at the surface of each proton and neutron is a quantum mechanically modeled interaction, and is the reason for the strong nuclear force. There is ongoing research on how to use the mathematical tool of lattice QCD for modeling the strong nuclear force.
So the answer to :
What is the real cause behind this strong force?
the strong colored force between elementary particles.
This has only removed one level of the onion. The next question "what are these forces between elementary particles" is the aim of the present generation of theoretical physicists, aiming for a Theory Of Everything. If we ever reach a TOE , then we will have the ultimate answers on why and what questions in physics. At present we end up in saying : because that is what fits the data and is predictive.