How does the exchange of pions result in the strong force? I understand that the residual strong force is a result of an exchange of pions. But I fail to understand how this exchange results in a force that holds nuclei together! May this query please be answered?  
 A: It does not "result",it models the strong force

That short-range nucleon-nucleon interaction can be considered to be a residual color force extending outside the boundary of the proton or neutron. That strong interaction was modeled by Yukawa as involving an exchange of pions, and indeed the pion range calculation was helpful in developing our understanding of the strong force.
Once quantum electrodynamics had produced the picture of the electromagnetic force as a process of exchanging photons, the question of whether or not the other forces were also exchange forces was a natural one. In 1935, Hideki Yukawa reasoned that the electromagnetic force was infinite in range because the exchange particle was massless. He proposed that the short range strong force came about from the exchange of a massive particle which he called a meson. By observing that the effective range of the nuclear force was on the order of a fermi, a mass for the exchange particle could be predicted using the uncertainty principle. The predicted particle mass was about 100 MeV. It did not receive immediate attention since no one knew of a particle which fit that description.

.....

In 1947, Lattes, Muirhead, Occhialini and Powell conducted a high altitude experiment, flying photographic emulsions at 3000 meters. These emulsions revealed the pion, which met all the requirements of the Yukawa particle.
We now know that the pion is a meson, a composite particle, and the current view is that the strong interaction is an interaction between quarks, but the Yukawa theory stimulated a major advance in the understanding of the strong interaction.

So the model followed the successful electromagnetic models and proposed the pion was the carrier of the strong interaction, and was confirmed.
Lattice QCD is the tool to study strong forces and fairly recently this has been done for the Yukawa exchanges:

However, due to high complexities of the dynamics governing the quarks, the quantum Chromodynamics (QCD), it has been extremely difficult to study the origin of the strong nuclear force from QCD.
Very recently, Dr. N. Ishii (Univ. of Tsukuba), Dr. S. Aoki (Univ. of Tsukuba) and Dr. T. Hatsuda (Univ. of Tokyo) have succeeded, for the first time, in unraveling the nature of the nuclear force on the basis of lattice gauge theory originally formulated by Dr. K. Wilson (Nobel Prize Laureate, Physics, 1982). By carrying out massive numerical simulations using the IBM supercomputer "BlueGene/L" in High Energy Accelerator Research Organization (KEK) in Japan, they could prove not only the validity of the Yukawa’s meson theory from QCD but also the existence of a strong repulsive core of the nuclear force at short distance

For the feynman diagram that describes nucleon nucleon scattering using the yukawa potential see this.

How Does The Exchange Of Pions Result In The Strong Force?

In the same way that when one calculates the Feynman diagrams for the electromagnetic exchanges,  the Coulomb force appears, calculating the Feynman diagrams for pion exchange gives the strong nuclear force.
Force is dp/dt , the change in momentum, the Feynman diagrams organize this momentum transfer from the initial particles to the final, and this after the integration of the implied integrals, results in the force experienced in the interaction.
